Improving the Performance and Security of Multi-Hop Wireless Networks

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Title:
Improving the Performance and Security of Multi-Hop Wireless Networks
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1 online resource (264 p.)
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english
Creator:
Zhang,Chi
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University of Florida
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Gainesville, Fla.
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Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Electrical and Computer Engineering
Committee Chair:
Fang, Yuguang
Committee Members:
Wu, Dapeng
Khargonekar, Pramod
Chen, Shigang

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Subjects / Keywords:
network -- performance -- security -- wireless
Electrical and Computer Engineering -- Dissertations, Academic -- UF
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Electrical and Computer Engineering thesis, Ph.D.
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theses   ( marcgt )
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Abstract:
Multi-hop wireless networks (or wireless ad hoc networks) have been widely accepted as an indispensable component of next-generation communication systems to facilitate ubiquitous network access from anywhere at any time. Although offering significant benefits, they also provide unique research challenges over their wired counterparts. Of note are the issues associated with the design of efficient routing protocols and network security, etc. In this dissertation, we aim to address these challenging and fundamental issues in heterogeneous, large-scale multi-hop wireless networks, spanning mobile ad hoc networks, wireless sensor networks, and multi-hop cellular networks. Our contributions are mainly sixfold. First, for a wireless sensor network, we propose a coverage inference protocol which can provide the base station an accurate and in-time measurement of connected coverage with minimized overhead. Second, we consider decentralized routing problems in heterogeneous ad hoc networks, wherein each node is connected to all its neighbors within some fixed radius, as well as possessing random ling-range links to more distant nodes. We characterize the necessary and sufficient condition for greedy geographic routing to be efficient with nonhomogeneous node distribution. Third, we study the throughput-delay tradeoffs in mobile ad hoc networks with network coding, and compare results with the situation where only replication and forwarding are allowed in each node. Forth, we propose a novel and promising incentive paradigm called C4 to induce cooperative behaviors in multi-hop cellular networks with minimized overhead. Fifth, we study the relationships between varying trust metrics and trust-based routing protocols. By developing a formal model to describe different trust environment, we identify the basic algebraic properties that a trust metric must have in order to guarantee the correctness and optimality of different wireless routing protocols. Last, we studies the relationships between network security requirement and network performance degradation. We characterize the asymptotic behaviors of achievable secure throughput and delay when the network size is sufficiently large.
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Chi Zhang.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
Local:
Adviser: Fang, Yuguang.

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lcc - LD1780 2011
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UFE0042891:00001


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IMPROVINGTHEPERFORMANCEANDSECURITYOFMULTI-HOPWIRELESSNETWORKSByCHIZHANGADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011ChiZhang 2

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Toallwhonurturedmyintellectualcuriosity,academicinterests,andsenseofscholarshipthroughoutmylifetime,makingthismilestonepossible 3

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ACKNOWLEDGMENTS Firstandforemost,Iwouldliketoexpressmysinceregratitudetomyadvisor,Prof.YuguangFang,forhisinvaluableguidance,encouragementandsupportwithmyyearsinWirelessNetworksLaboratory(WINET).HeconvincedmetojointhePh.D.programatUFLsevenyearsago,andencouragedmeinmypursuitofanacademiccareeraftergraduation.Iamlookingforwardtoourcollaborationinthefuture.IalsowouldliketothankProfessorPramodKhargonekar,ProfessorDapengWu,andProfessorShigangChenforservingonmysupervisorycommitteeandfortheirgreathelpinvariousstagesofmyworkandcareer.Iwouldnotbeasanegraduatestudentwithoutagroupofgreatfriends.IwouldliketoextendmythankstoallmycolleaguesinWINETforprovidingmeawarm,family-likeenvironmentandfortheircollaborationandinsightfuladvice.IspeciallythankDr.WenjingLou,Dr.WenchaoMa,Dr.Byung-SeoKim,Dr.WeiLiu,Dr.XiangChen,Dr.JingZhao,Dr.HongqiangZhai,Dr.YanchaoZhang,Dr.ShushanWen,Dr.JianfengWang,Dr.YunZhou,Dr.XiaoxiaHuang,Dr.PanLi,Dr.FengChen,Dr.JinyuanSun,Dr.YangSong,Dr.RongshengHuang,FrankGoergen,MiaoPan,HaoYue,LinkeGuo,HuangLin,YuanxiongGuo,ZongruiDing,XinxinLiu,Dr.SunmyengKim,Dr.NicolaScalabrino,Dr.RobertoRiggio,Dr.QiangShen,Dr.ZhiqiangShi,Dr.XiaoyanYin,Dr.XiaobinTan,Dr.GuoliangYaoandDr.XihuaDongformanyvaluablediscussionsandallthegoodmemories.SpecialthanksareduetotheFangfamily:myadvisorYuguangFangandhiswifeJenniferLu,andtotheHuangfamily:RongshengandHaiyan,whonotonlyconstantlyencouragedmeandhelpedmeinmanyways,butalsosharedtheirviewoflifewithme.Finally,Ioweaspecialdebtofgratitudetomybelovedparentsandsister.Withouttheirloveandunwaveringsupport,IwouldneverimaginewhatIhaveachieved. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 10 LISTOFFIGURES ..................................... 11 ABSTRACT ......................................... 14 CHAPTER 1INTRODUCTION ................................... 16 1.1WirelessAdHocNetworks:AnOverview .................. 16 1.2ResearchChallenges ............................. 18 1.2.1Scalability ................................ 18 1.2.2Heterogeneity .............................. 19 1.2.3Client-ServerModelShift ....................... 19 1.2.4Security ................................. 19 1.3ScopeandOrganizationoftheDissertation ................. 20 1.3.1ChaptersonNetworkPerformance .................. 20 1.3.2ChaptersonNetworkSecurity ..................... 21 2DESIGNINGCOVERAGEINFERENCEPROTOCOLSFORWSNS ...... 23 2.1ChapterOverview ............................... 23 2.2Preliminaries .................................. 26 2.2.1NetworkModel ............................. 26 2.2.2DesignGoals .............................. 27 2.3BOND:BoundaryNodeDetectionScheme ................. 28 2.3.1BoundaryNodeandItsDetectionAlgorithm ............. 28 2.3.2LocalizedVoronoiPolygons ...................... 31 2.3.3LVP-BasedBoundaryNodeDetection ................ 33 2.3.3.1Input ............................. 33 2.3.3.2Algorithm ........................... 34 2.3.3.3Output ............................ 35 2.3.4AlgorithmValidation .......................... 35 2.3.5DiscussionsonBOND ......................... 37 2.3.6LocalityofBoundaryNodeDetection ................. 39 2.4CIP:CoverageInferenceProtocol ...................... 40 2.4.1NeighborhoodMonitoringandSelf-Detection ............ 41 2.4.2Self-ReportingofBoundaryNodes .................. 43 2.4.3ExplicitACKsfromtheBS ....................... 43 2.5ComparisonandSimulation .......................... 46 2.5.1Boundary-Node-BasedApproaches ................. 47 5

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2.5.1.1Polygon-basedschemes .................. 47 2.5.1.2Perimeter-basedschemes ................. 51 2.5.2Aggregation-BasedApproaches .................... 54 2.5.2.1Naiveschemes ........................ 54 2.5.2.2Spatialaggregation-basedschemes(SAB) ........ 55 2.6ExtensionstoCIP ............................... 58 2.6.1Location-Error-TolerantCIP ...................... 58 2.6.2Prediction-BasedCIP ......................... 59 2.7ChapterSummary ............................... 60 3LINKHETEROGENEITYANDDECENTRALIZEDROUTING .......... 62 3.1ChapterOverview ............................... 62 3.2RelatedWork .................................. 64 3.2.1RelatedWorkonSocialNetworks ................... 64 3.2.2RelatedWorkonWirelessNetworks ................. 67 3.3NetworkModel ................................. 68 3.3.1NotationandNetworkModel ...................... 68 3.3.2Background ............................... 72 3.4CharacterizationoftheParametersintheNetworkModel ......... 73 3.4.1NormalizationParameteranandtheExpectedNumberofShortcutsforEachNode .............................. 73 3.4.2GEOGREEDYParametercgandtheExpectedNumberofLocalNeighborsforEachNode ....................... 74 3.5NavigabilityofNonhomogeneousPoissonNetworks ............ 77 3.5.1NavigabilityofNPN(n,rn,1) ...................... 79 3.5.2InnavigabilityofNPN(n,rn,)When6=1 .............. 82 3.6ApplicationstoWirelessAdHocNetworks .................. 84 3.6.1DoestheDistributionofShortcutsCount? .............. 84 3.6.2DoesAddingMoreShortcutsHelp? .................. 87 3.7ChapterSummary ............................... 88 4SCALINGLAWSFORLARGE-SCALEMANETSWITHNETWORKCODING 89 4.1ChapterOverview ............................... 89 4.2BackgroundandRelatedWork ........................ 92 4.2.1ScalingLawsofMANETswithoutNetworkCoding ......... 92 4.2.2NetworkCodingApplicationsinWirelessNetworks ......... 94 4.2.3ScalingLawsofWirelessNetworkswithNetworkCoding ...... 95 4.3MANETModelsandDenitions ........................ 96 4.3.1NetworkModels ............................. 96 4.3.2NetworkPerformanceMetrics ..................... 99 4.4Throughput-Delay-StorageTradeoffswithoutNetworkCoding ....... 101 4.4.1Throughput-DelayTradeoffswithInniteBufferSpaces ....... 101 4.4.2Throughput-StorageTradeoffs ..................... 103 6

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4.5Throughput-Delay-StorageTradeoffswithNetworkCoding:SchemesandResults ................................... 104 4.5.1NetworkCodingOperation ....................... 104 4.5.2RLC-BasedRelaySchemes ...................... 106 4.5.3MainResultsaboutRLC-BasedSchemes .............. 109 4.6Throughput-Delay-StorageTradeoffswithNetworkCoding:Analysis ... 112 4.6.1Preliminaries .............................. 112 4.6.2Performanceof2-HopRelaywithRLC ................ 116 4.6.3PerformanceofMulti-HopRelaywithRLC .............. 118 4.7ChapterSummary ............................... 120 5PROVIDINGINCENTIVESINMULTI-HOPWIRELESSNETWORKSWITHNETWORKCODING ................................. 121 5.1ChapterOverview ............................... 121 5.2RelatedWorkandMotivationforC4 ..................... 124 5.2.1ExistingIncentiveMechanismsforMWNs .............. 124 5.2.2MotivationforOurC4 .......................... 128 5.2.3NetworkCodingandIncentives .................... 130 5.3DesignandImplementationofOurC4 .................... 131 5.3.1SystemModelandProblemFormulation ............... 131 5.3.2MethodologyofOurC4 ........................ 132 5.3.3ImplementationDetailsofOurC4 ................... 135 5.4PerformanceAnalysisofOurC4 ....................... 138 5.4.1NetworkModelforPerformanceAnalysis ............... 138 5.4.2PerformanceAnalysisforBroadcastandMulticastTrafcs ..... 140 5.4.3PerformanceAnalysisforPureUnicastTrafcs ........... 145 5.5ImprovingOurC4'sPerformancewithSocialContactInformation ..... 149 5.5.1InformationHighwayandMulti-hopRelay .............. 150 5.5.2CommunityStructureandGroupingParameterSelection ...... 152 5.6ChapterSummary ............................... 155 6TRUST-BASEDROUTINGANDNON-CLASSICALROUTINGALGEBRA ... 157 6.1ChapterOverview ............................... 157 6.2MotivatingExamples:WhyDoWeNeedaFormalStudy? ......... 162 6.3AbstractFrameworkforTrustMetricsandTrust-BasedRouting ...... 168 6.3.1SystemModelforTrustManagement ................. 168 6.3.2GraphModelsforWANETs ...................... 170 6.3.3FormalizingTrustMetricSpace .................... 171 6.3.4FormalizingRoutingProtocols ..................... 173 6.4PathAlgebraforIndirectTrustInference ................... 177 6.4.1AlgebraicFoundations ......................... 177 6.4.2TrustInferenceProblemFormalization ................ 178 6.4.3VericationoftheBi-MonoidProperties ............... 180 6.4.4SolvingPathAlgebraicProblems ................... 181 7

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6.5RoutingAlgebraforUniformTrustEnvironment ............... 183 6.5.1Non-ClassicRoutingAlgebraforTrust-BasedRouting ....... 184 6.5.2ConditionsforCorrectandOptimalRouting ............. 186 6.5.3IllustratingExamples .......................... 187 6.6RoutingAlgebraforGroup-BasedTrustEnvironment ............ 189 6.6.1MotivatingExample ........................... 189 6.6.2ProblemFormalization ......................... 191 6.6.3PropertiesofConversionFunctions .................. 193 6.6.4ConditionsforCorrectandOptimalRouting ............. 195 6.7ChapterSummary ............................... 197 7SECURENETWORKPERFORMANCEOFLARGE-SCALEWANETS ..... 199 7.1ChapterOverview ............................... 199 7.2BackgroundandRelatedWork ........................ 203 7.2.1OnPre-DistributionofKeyingMaterials/SAs ............. 203 7.2.2OnSecureConnectivity ........................ 205 7.2.3OnSecureThroughput ......................... 206 7.3SystemAssumptionsandMainResults ................... 208 7.3.1RandomNetworkModelofWANETs ................. 208 7.3.1.1Nodedistribution ....................... 208 7.3.1.2Interferencemodels ..................... 209 7.3.1.3Trafcpattern ......................... 210 7.3.2NetworkPerformanceMetrics ..................... 211 7.3.2.1(Secure)throughput ..................... 211 7.3.2.2(Secure)delay ........................ 211 7.3.2.3Thepriceforsecurity .................... 211 7.3.3MainResultsofOurWork ....................... 211 7.4NetworkPerformancewithoutSLA ...................... 214 7.4.1SchemeDescription .......................... 214 7.4.2PerformanceAnalysisofScheme1 .................. 217 7.5NetworkPerformancewithSLA ........................ 220 7.5.1SchemeDescription .......................... 220 7.5.2PerformanceAnalysisofScheme3 .................. 225 7.6OptimalityofOurSchemes .......................... 227 7.6.1UpperBoundsonSecureThroughputs ................ 227 7.6.2LowerBoundsonSecureDelays ................... 229 7.7ChapterSummary ............................... 229 8CONCLUSIONANDFUTUREDIRECTIONS ................... 230 8.1DissertationSummary ............................. 230 8.2FutureDirections ................................ 234 APPENDIX ANETWORKTOPOLOGIESUSEDINPERFORMANCEEVALUATIONS .... 238 8

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BASYMPTOTICNOTATION .............................. 241 CSOMERESULTSABOUTTORUSPARTITIONSINSCHEME1AND3' .... 242 DSECURENETWORKPERFORMANCEUNDERTHEPHYSICALMODEL .. 247 REFERENCES ....................................... 250 BIOGRAPHICALSKETCH ................................ 264 9

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LISTOFTABLES Table page 4-1Networkperformancesunderfastmobilitymodel ................. 102 4-2Networkperformancesunderslowmobilitymodel ................ 103 5-1Thedesignspaceofincentivemechanisms .................... 122 5-2Costanalysisofdifferentincentivemechanisms .................. 129 5-3Examplesofmulti-hopwirelessnetworks ..................... 133 5-4Datasetproperties .................................. 149 6-1NodelabelsinFigure 6-7 .............................. 175 10

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LISTOFFIGURES Figure page 1-1Examplesofwirelessnetworkswithinfrastructures ................ 16 1-2Exampleofawirelessadhocnetwork ....................... 17 2-1AnexemplaryWSNs ................................. 29 2-2IllustrationoftheLVP-basedboundarynodedetectionalgorithm ........ 31 2-3IllustrationoftheproofofTheorem 2.1 ....................... 37 2-4Non-localityoftheboundarynodedetectionwhenrc<2rs ............ 39 2-5BasicoperationsoftheBOND-basedCIP ..................... 40 2-6EnergyconsumptionfortheLVP-andVP-basedschemes ........... 48 2-7Perimeter-basedboundarynodedetectionapproaches ............. 50 2-8Averagenumberofneighbornodesneededforthecrossing-coveragecheckingapproachandourBOND .............................. 52 2-9SimulationresultswithTEUP=10s,Pr[FA]0.01 ................ 53 2-10Simulationresultswithresponsedelay40s,Pr[FA]0.01 ........... 53 2-11IllustrationofSAB .................................. 55 2-12EnergycostratioofSABandCIPtonaivescheme ................ 57 2-13Voronoi-diagrambasedcoverageholeprediction ................. 59 3-1SnapshotsondensityofwirelessusersinRomeCityon30August2006 .... 65 3-2Navigablesmall-worldnetworkmodels ....................... 72 3-3Sufcientconditionforualwayshavingalocalneighborwclosertothedestinationvwithd(u,v)>rn .................................. 75 3-4ApproximateGEOGREEDYroutingalgorithm .................... 78 3-5CalculatingtheprobabilityofnodeuhavingashortcuttooneofthenodesinA(t,d=2) ....................................... 79 4-1FastandslowmobilitymodelsforMANETs .................... 97 4-2Celltransmissionscheduling ............................ 99 4-3TimetablesfordifferentRLC-basedschemesunderslowmobilitymodel .... 110 11

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5-1Trademodelsforelementaryinteractionsbetweentwonodes .......... 125 5-2Agenericarchitectureformulti-hopwirelessnetworks .............. 132 5-3Acomparisonbetweenbartering(withoutcoding)andourC4 .......... 134 5-4PacketformatinourC4 ............................... 137 5-5Statetransitiondiagramforobtainingpacketsofamobilenode ......... 142 5-6TD(IM)andCP(IM)asfunctionsofK ....................... 144 5-7Theeffectiveness-costtradeoffsofourC4forpureunicasts ........... 148 5-8Informationhighwayandmulti-hoprelay ...................... 151 5-9Thedistributionofhopcounts ............................ 151 5-10Communitystructuresinthesocialcontactgraph ................. 153 5-11Theeffectiveness-costtradeoffsunderdifferentg(ork)values ......... 154 6-1PhysicalgraphandtrustgraphforanexemplarypathfromAtoE ........ 160 6-2Diversityoftrustmetrics ............................... 162 6-3Algebraicpathformulationforindirecttrustinferenceproblems ......... 164 6-4Distributivityoftrustmetrics ............................. 165 6-5Examplesofroutinganomaliesintrust-basedrouting ............... 166 6-6Systemmodelfortrustmanagementinanycommunicationsystemsandtrust-basedrouting ......................................... 168 6-7Exemplaryin-treesforrootnodev0(destination) ................. 174 6-8Flowgraphs,functionandtrustevaluationonpaths ............... 185 6-9Flowgraphsforthephysicalpathsfromnodev1tonodev0inExample5inSection 6.2 ...................................... 188 6-10CalculatingfunctionforthephysicalpathgiveninFigure 6-9 (c) ........ 189 6-11Trustinmultiplegroups ............................... 190 6-12Concatenationofpathsfromdifferentgroups ................... 192 7-1Impactofsecurityrequirementsonthroughputscalinginrandomnetworks .. 212 7-2ThesecurecommunicationschemewithoutSLA ................. 214 7-3Multi-hopSLAoperations .............................. 221 12

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7-4Securecommunicationscheme3`withSLA .................... 222 7-5Routingschemeonthepercolatedgrid ...................... 223 A-1Somenetworktopologiesusedinourperformanceevaluation .......... 239 C-1Cellschedulingscheme ............................... 246 13

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyIMPROVINGTHEPERFORMANCEANDSECURITYOFMULTI-HOPWIRELESSNETWORKSByChiZhangAugust2011Chair:YuguangFangMajor:ElectricalandComputerEngineeringMulti-hopwirelessnetworks(orwirelessadhocnetworks)havebeenwidelyacceptedasanindispensablecomponentofnext-generationcommunicationsystemstofacilitateubiquitousnetworkaccessfromanywhereatanytime.Althoughofferingsignicantbenets,theyalsoprovideuniqueresearchchallengesovertheirwiredcounterparts.Ofnotearetheissuesassociatedwiththedesignofefcientroutingprotocolsandnetworksecurity,etc.Inthisdissertation,weaimtoaddressthesechallengingandfundamentalissuesinheterogeneous,large-scalemulti-hopwirelessnetworks,spanningmobileadhocnetworks,wirelesssensornetworks,andmulti-hopcellularnetworks.Ourcontributionsaremainlysixfold.First,forawirelesssensornetwork,weproposeacoverageinferenceprotocolwhichcanprovidethebasestationanaccurateandin-timemeasurementofconnectedcoveragewithminimizedoverhead.Second,weconsiderdecentralizedroutingproblemsinheterogeneousadhocnetworks,whereineachnodeisconnectedtoallitsneighborswithinsomexedradius,aswellaspossessingrandomling-rangelinkstomoredistantnodes.Wecharacterizethenecessaryandsufcientconditionforgreedygeographicroutingtobeefcientwithnonhomogeneousnodedistribution.Third,westudythethroughput-delaytradeoffsinmobileadhocnetworkswithnetworkcoding,andcompareresultswiththesituationwhereonlyreplicationandforwardingareallowedineachnode.Forth,weproposeanovelandpromisingincentiveparadigmcalledC4to 14

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inducecooperativebehaviorsinmulti-hopcellularnetworkswithminimizedoverhead.Fifth,westudytherelationshipsbetweenvaryingtrustmetricsandtrust-basedroutingprotocols.Bydevelopingaformalmodeltodescribedifferenttrustenvironment,weidentifythebasicalgebraicpropertiesthatatrustmetricmusthaveinordertoguaranteethecorrectnessandoptimalityofdifferentwirelessroutingprotocols.Last,westudiestherelationshipsbetweennetworksecurityrequirementandnetworkperformancedegradation.Wecharacterizetheasymptoticbehaviorsofachievablesecurethroughputanddelaywhenthenetworksizeissufcientlylarge. 15

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CHAPTER1INTRODUCTION 1.1WirelessAdHocNetworks:AnOverviewRecentyearshavewitnessedasurgeofresearchanddevelopmentforwirelessadhocnetworks(orwirelessmulti-hopnetworks)astheyhavetremendousmilitaryandcommercialpotential.Awirelessadhocnetwork(WANET)[ 119 131 ]isawirelessnetwork,comprisedofmobilecomputingdevicesthatusewirelesstransmissionforcommunication,havingnoxedinfrastructure(acentraladministrationsuchasabasestationinacellularwirelessnetworkoranaccesspointinawirelesslocalareanetwork,cf.Figure 1-1 ).Themobiledevicesalsoserveasroutersduetothelimitedrangeofwirelesstransmissionofthesedevices,thatis,severaldevicesmayneedtorouteorrelayapacketbeforeitreachesitsnaldestination(cf.Figure 1-2 ).Adhocwirelessnetworkscanbedeployedquicklyanywhereandanytimeastheyeliminatethecomplexityofinfrastructuresetup.Thesenetworksndapplicationsinseveralareas.Someoftheseinclude:militarycommunications(establishingcommunicationamongagroupofsoldiersfortacticaloperationswhensettingupaxedwirelesscommunicationinfrastructureinenemyterritoriesorininhospitableterrainsmaynotbepossible),emergencysystems(forexample,establishingcommunicationamongrescuepersonnelindisaster-affectedareas)thatneedquickdeploymentofanetwork,collaborativeanddistributedcomputing,andhybrid(integratedcellularandadhoc)wirelessnetworks. Figure1-1. Examplesofwirelessnetworkswithinfrastructures. 16

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Figure1-2. Exampleofawirelessadhocnetwork(wirelessnetworkwithoutinfrastructure). Ingeneral,wirelessadhocnetworkscanbeclassiedintotwocategories,mobileadhocnetworks(MANETs)andstaticadhocnetworks.Theformercomprisenetworknodesthatarefreetomoveaboutrandomlyandorganizethemselvesarbitrarily.ExemplaryapplicationscenariosofMANETsincludetacticalmilitaryoperations,homelandsecurity,emergencydisasterreliefandrescue,andsoon.Mostrecently,MANETshavebeenextendedtogeneralciviliancontextsandareoftenreferredtoaswirelessmeshnetworks[ 5 ],wheremobileuserscanaccessthenetworkeitherthroughadirectwirelesslinktoawirelessaccesspoint(AP),orthroughasequenceofintermediateuserstoanAPthatistoofarawaytoreach.Bycontrast,staticadhocnetworksmainlyconsistofstationarynodes,thatis,xedatwheretheyweredeployed.Themostsignicantexampleofthislatertypeiswirelesssensornetworks[ 4 ],whichhaveattractedextensiveattentioninbothacademiaandindustryfortheirbroadpotentialinnotonlymilitaryandhomelandsecurityscenariosbutalsoingeneralciviliansettings.Themostfundamentalfunctionalityofanywirelessadhocnetworkistoprovideend-to-endcommunicationinapeer-to-peer(P2P)fashion,i.e.,withoutanyinfrastructure(orwithpartialinfrastructure).Figure 1-2 depictsthepeer-levelmulti-hoprepresentationofsuchanetwork.MobilenodeAcommunicateswithanothersuchnodeBdirectly(single-hop)wheneveraradiochannelwithadequatepropagationcharacteristicsisavailablebetweenthem.Otherwise,multi-hopcommunicationisnecessarywhereoneormoreintermediatenodesmustactasarelay(router)betweenthecommunicatingnodes.Forexample,thereisnodirectradiochannel(shownbythelines)betweenA 17

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andCorAandGinFigure 1-2 .NodesBandFmust,therefore,serveasintermediateroutersforcommunicationbetweenAandC,andAandG.Indeed,adistinguishingfeatureofadhocnetworksisthatallnodesmustbeabletofunctionasroutersondemand.Topreventpacketsfromtraversinginnitelylongpaths,anobviousessentialrequirementforchoosingapathisthatthepathmustbeloop-free.Aloop-freepathbetweenapairofnodesiscalledaroute. 1.2ResearchChallengesWhileofferingsignicantbenets,wirelessadhocnetworksalsoprovideuniqueresearchchallengesovertheirwiredcounterparts.Thissubsectionoutlinesthemajorproblemsthatoughttobeaddressed.Theprotocoldependentdevelopmentpossibilitiesaremostlyomittedandthefocusisonthebigpicture,ontheproblemsthatstandinawayofhavingpeer-to-peerconnectivityeverywhereinthefuture. 1.2.1ScalabilityMostofthevisionariesdepictingapplicationswhichareanticipatedtobenetfromtheadhocandsensornetworkingtechnologytakescalabilityasgranted.Imagine,forexample,thevisionofubiquitouscomputingwherenetworkscanbeofanysize.However,itisunclearhowsuchlargenetworkscanactuallygrow.Adhocnetworkssuffer,bynature,fromthescalabilityproblemsincapacity.Toexemplifythis,wemaylookintosimpleinterferencestudies.Inanon-cooperativenetwork,whereomni-directionaLantennasarebeingused,thethroughputpernodedecreasesatarate1=p n,wherenisthenumberofnodes[ 60 ].Thatis,inanetworkwith100nodes,asingledevicegets,atmost,approximatelyonetenthofthetheoreticalnetworkdatarate.Thisproblem,however,cannotbexedexceptbyphysicallayerimprovements,suchasdirectionalantennas.Iftheavailablecapacitysetssomelimitsforcommunications,sodotheprotocols.Routeacquisition,servicelocationandencryptionkeyexchangesarejustfewexamplesoftasksthatwillrequireconsiderableoverheadasthenetworksizegrows.Ifthescarce 18

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resourcesarewastedwithprofusecontroltrafc,thesenetworksmayseeneverthedaydawn.Therefore,scalabilityisacrucialresearchtopicandhastobetakenintoaccountinthedesignofsolutionsforadhocandsensornetworks. 1.2.2HeterogeneityNetworkheterogeneityisacertaintyfortoday'swirelessnetworks.Therearetwokindsofheterogeneity:rst,thedistributionofwirelessusers/devicesinthephysicalspaceisnon-homogeneous;second,wirelessdevicesarelikelytohavewidelyvaryingradioranges(e.g.,cellular/WiMax,WiFi,Zigbee).Foraheterogeneousadhocnetwork,whereshort-rangewirelesslinksandlong-rangewirelesslinks(shortcuts)coexist,howtodesignefcientdecentralizedroutingprotocolswithlocalinformationisanopenproblemintheliterature. 1.2.3Client-ServerModelShiftIntheInternet,anetworkclientistypicallyconguredtouseaserverasitspartnerfornetworktransactions.Theseserverscanbefoundautomaticallyorbystaticconguration.Inadhocnetworks,however,thenetworkstructurecannotbedenedbycollectingIPaddressesintosubnets.Theremaynotbeservers,butthedemandforbasicservicesstillexists.Addressallocation,nameresolution,authenticationandtheservicelocationitselfarejustexamplesoftheverybasicserviceswhichareneededbuttheirlocationinthenetworkisunknownandpossiblyevenchangingovertime.Duetotheinfrastructurelessnatureofthesenetworksandnodemobility,adifferentaddressingapproachmayberequired.Inaddition,itisstillnotclearwhowillberesponsibleformanagingvariousnetworkservices.Therefore,whiletherehasbeenvastresearchinitiativesinthisarea,theissueofshiftfromthetraditionalclient-severmodelremainstobeappropriatelyaddressed. 1.2.4SecurityWirelessadhocnetworksareparticularlypronetomaliciousbehavior.Lackofanycentralizednetworkmanagementorcerticationauthoritymakesthesedynamically 19

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changingwirelessstructuresveryvulnerabletoinltration,eavesdropping,interference,andsoon.Securityisoftenconsideredtobethemajorroadblockinthecommercialapplicationthistechnology.Securityisindeedoneofthemostdifcultproblemstobesolved.Someofthemajorsecuritychallengesthatawirelessadhocnetworkfacesincludethefollowing: Alloldthreatstoaconventionalwirednetworkapplytoawirelessadhocnetwork. Thesharedwirelessmediumfacilitatespassiveeavesdroppingondatacommunicationsand/oractivebogusmessageinjectionintothenetworkbyattackers. Earlyprotocoldesignforwirelessadhocnetworksallassumedafriendlyandcooperativeenvironment.Assuch,manywirelessprotocolshaveinherentsecurityaws. Mobiledevicesaresubjecttophysicaltheftorloss,leadingtoinsiderattackslaunchedbyattackersharnessingcondentialinformationextractedfromstolendevices. Intrusiondetectionisfarmoredifcult,mainlybecauseitishardtodifferentiateanomaliescausedbycharacteristicsofwirelesschannelsandthosecausedbyattacks. Thereisoftenlackofanon-linecentralizedauthorityoradministration. Mobiledevicesusuallyhavestringentresourceconstraintsandthuscannotaffordresource-hungrysecurityprotocols. 1.3ScopeandOrganizationoftheDissertationThisdissertationcontributestothedevelopmentofnovelsolutionstoanumberofchallengingandfundamentalissuesinwirelessmulti-hopnetworks(orwirelessadhocnetworks),whichareeitherignoredornotwelladdressedinpreviousresearch.Therestofthedissertationcanbedividedintotwomainpartsasfollows. 1.3.1ChaptersonNetworkPerformanceFromChapter 2 toChapter 4 ,wewilldiscussseveraltechniquestoimprovenetworkperformances,likecoverage,throughput,anddelay. 20

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Connectedcoverage,whichreectshowwellatargeteldismonitoredunderthebasestation(BS),isthemostimportantperformancemetricsusedtomeasurethequalityofsurveillancethatwirelesssensornetworkscanprovide.InChapter 2 ,weproposeacoverageinferenceprotocol(CIP)whichcanprovidetheBSanaccurateandin-timemeasurementChapter 3 considersdecentralizedroutingproblemsinheterogeneousadhocnetworks,whereineachnodeisconnectedtoallitsneighborswithinsomexedradius,aswellaspossessingrandomshortcutstomoredistantnodes.Weshowthatwithnonhomogeneousnodedistribution,thenecessaryandsufcientconditionforgreedygeographicroutingtobeefcientisthattheprobabilityofashortcutbeingpresentfromnodeutovshouldbeinverselyproportionaltothenumberofnodeswhichareclosertouthanvis.Tothebestofourknowledge,thisistherstworktoprovethisresultinthenonhomogeneouscontinuumsetting.Chapter 4 characterizesthethroughput-delaytradeoffsinmobileadhocnetworks(MANETs)withnetworkcoding,andcomparesresultswiththesituationwhereonlyreplicationandforwardingareallowedineachnode.Theschemes/protocolsachievingthosetradeoffsinaneffectiveanddecentralizedwayareproposedandtheoptimalityofthosetradeoffsisestablished. 1.3.2ChaptersonNetworkSecurityWewillconcentratenetworksecurityproblemsinthefollowingthreechapters.Foramulti-hopwirelessnetwork(MWN)consistingofmobilenodescontrolledbyindependentself-interestedusers,incentivemechanismisessentialformotivatingmobilenodestocooperateandforwardpacketsforeachother.InChapter 5 ,weproposeanovelandpromisingincentiveparadigm,ControlledCodedpacketsasvirtualCommodityCurrency(C4),toinducecooperativebehaviorsinMWNs.InourC4,throughintroducingseveraltechniquesfromnetworkcoding,codedinformationpacketsareutilizedasanewkindofvirtualcurrencytofacilitatepacket/serviceexchanges 21

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amongself-interestednodesinaMWN.Sincethevirtualcurrencyimplementedinthiswayalsocarriesusefuldatainformation,itisthecounterpartoftheso-calledcommoditycurrencyinthephysicalworld,andtheoverheadbroughtbyC4isextremelysmallcomparedtotraditionalschemes.Chapter 6 studiestherelationshipsbetweentrustmetricsandtrust-basedroutingprotocols.Bydevelopingaformalmodelstodescribedifferenttrustenvironment,weidentifythebasicalgebraicpropertiesthatatrustmetricmusthaveinordertoguaranteethecorrectnessandoptimalityofdifferentgeneralizeddistance-vectororlink-stateroutingprotocolsinwirelessadhocnetworks.Theproposedresearchprovidesanewmethodologyfortheformalanalysisofwirelessnetworksecurityandacceleratestheevaluation,designandrealdeploymentoftrust-basedroutingprotocols.Securityalwayscomeswithapriceintermsofperformancedegradation,whichshouldbecarefullyquantied.Thisisespeciallythecaseforwirelessadhocnetworkswhichoffercommunicationsoverasharedwirelesschannelwithoutanypre-existinginfrastructure.InChapter 7 ,basedonageneralrandomnetworkmodel,theasymptoticbehaviorsofsecurethroughputanddelaywiththecommontransmissionrangernandtheprobabilitypfofneighboringnodeshavingaprimarysecurityassociationarequantiedwhenthenetworksizenissufcientlylarge.Finally,Chapter 8 summarizesthisdissertationandpointsoutsomefutureresearchdirections. 22

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CHAPTER2DESIGNINGCOVERAGEINFERENCEPROTOCOLSFORWSNS 2.1ChapterOverviewAwirelesssensornetwork(WSN)isalargecollectionofdenselydeployed,spatiallydistributed,autonomousdevices(ornodes)thatcommunicateviawirelessandcooperativelymonitorphysicalorenvironmentalconditions[ 4 79 ].Insuchnetwork,sensornodesaredeployedoverageographicarea(calledtheregionofinterestorROI)byaerialscatteringorothermeans.Eachsensornodecanonlydetecteventswithinsomeverylimiteddistancefromitself,calledthesensingrange.Inaddition,sensornodesnormallyhavefairlylimitedtransmissionandreceptioncapabilitiessothatsensingdatahavetoberelayedviaamulti-hoppathtoadistantbasestation(BS),whichisadatacollectioncenterwithsufcientlypowerfulprocessingcapabilitiesandresources.Afterbeingdeployed,sensornodesareusuallyleftunattendedforaverylongperiodoftimesothattheymayfailovertimeduetovariousreasonssuchasbatteryexhaustionandphysicaldestructionsbyattackers.Theymayalsobemovedawayfromwheretheyweredeployedbyanimals,winds,orotherenvironmentalmeans.Asaconsequenceofnodefailures,nodemovementsandotherunpredictablefactors,thenetworktopologymaychangewithtime.Itis,therefore,criticalthattheBSlearninrealtimehowwelltheWSNperformsthegivensensingtaskunderdynamicallychangingnetworktopology.FromtheBS's(oruser's)pointofview,apositionintheROIisreallyunderthesurveillanceoftheWSNifandonlyifthispositioniswithinthesensingrangeofatleastonesensornodeconnectedtotheBS.WedenethecollectionofallthesepositionsintheROIastheconnectedcoverage(orcoverageinshort).Obviously,itisoneofthemostimportantperformancemetricmeasuringthequalityofsurveillanceaWSNcanprovide.TheBSalsoshouldhavetheabilitytomonitorthecoveragestatusinrealtime. 23

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Althoughmuchresearch[ 10 40 45 59 69 92 93 ]hasbeenconductedtoensurehighnetworkcoverageandconnectivityfortheWSN,noneofthemaddresseshowtohelptheBSinferthecoverageboundarywhencoverageholesemerge.Possiblecausesleadingtocoverageholesincludeenergydepletionofsensornodes,intendedattacksonsensornodes,andsoon.InmanyWSNapplications,especiallysecurity-sensitiveapplications,itisamusttoaccuratelydetectthecoverageboundary.Theprotocoldevelopedinthischaptercanafrmativelyanswerthisopenchallengingissue.Ontheotherhand,problemsrelatedtotheself-monitoringofaWSNhavebeenstudiedintheliteratureforvariousapplicationsandpurposes.Forexample,ChessaandSanti[ 137 ]proposeasingletime-outschemetomonitorthesystem-levelfaultdiagnosis.In[ 179 ],aresidualenergyscanisdesignedtoapproximatelydepicttheremainingenergydistributionwithinaWSN.Inaddition,aself-monitoringmechanismfordetectingnodefailuresisproposedin[ 68 ].However,Alltheseschemescannotbedirectlyusedforthecoverageinference,astheyareeithercentralizedschemesorassumethateachindividualsensorintheWSNneedstobemonitored.ThisisnottrueforourcasebecausetheBSonlyneedstoensurethataceratinpercentageofthesensorsarefunctioning,especiallywhentheWSNisdenselydeployed.Generally,wecandistinguishtwobasictypesofWSNs:proactiveandreactive.ProactiveWSNsinvolveaperiodicdatadeliverybetweensensornodesandtheBSs.Bycomparison,inreactiveWSNs,packetsaresentonlywhensomeeventofinterestoccursandissensed.AlthoughforproactiveWSNs,eachnodecansimplyhelptheBSinfertheconnectedcoveragebypiggybackingitsstatusinformationondatatrafc,itiswellknownthatproactiveWSNsareenergyinefcientandnotscalable[ 21 79 ].Therefore,inthischapter,wefocusonprovidingcoverageinferencefortheBSinreactiveWSNs.However,duetotheverynatureofWSNs,thistaskisfarfromtriviality.Themostsignicantchallengeisduetothestrictresourcelimitationofsensornodes(batterypower,memory,computationalcapability,etc.)whichhighlightstheneedfor 24

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alocalizedsolution.Althoughthereisplentyofwork[ 40 53 62 69 157 174 ]onthelocalizedcoverageboundarydetection,noneofthemisadequatebecausesomeofthemsuchaspolygon-basedapproaches[ 40 53 157 ]arenottrulylocalizedsolutionsandsufferfrompossiblysignicantcommunicationoverhead,whileotherssuchasperimeter-basedschemes[ 62 69 174 ]areinefcientwhenthenodedensityishigh(cf.Section 2.5 ).Generalizationofsuchschemestoallthesituationsweareinterestedinisnottrivial.Thischaptermakesthefollowingcontributions.First,wepresentaCoverageInferenceProtocol(CIP)whichcanprovidetheBSanaccurateandin-timemeasurementofthecurrentconnectedcoverageinanenergy-efcientway.Second,weshowthatthemajorcomponentofourCIP-BOundaryNodeDetection(BOND)scheme-canbereusedtoprovidemanyotherfunctionalitiesforWSNs,suchastopologycontrol,efcientrouting,sleepingscheduling,andspatialaggregation.Therefore,ourschemescanbeexploitedtoseamlesslyintegratemultiplefunctionalitieswithlowoverhead.TheperformanceofourBOND-basedCIPcomparedwithotherpossibleCIPsisalsoinvestigated.Moreover,wedeviseextensionstoCIPwhichcantoleratelocationerrorsandactivelypredictthechangeoftheconnectedcoveragebyutilizingtheinformationofresidualenergyinformationofsensornodes.Therestofthischapterisorganizedasfollows.Section 2.2 introducesthenetworkmodelandthedesigngoalsofourCIP.NextwedetailthecorecomponentofoursolutioncalledBONDinSection 2.3 .ThisisfollowedbypresentingthecompleteCIPinSection 2.4 anditscomparisonwithotherpossiblealternativesinthedesignspaceinSection 2.5 .WethenpresentseveralextensionstoCIPinSection 2.6 andendwithconclusionsandfuturework. 25

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2.2Preliminaries 2.2.1NetworkModelThroughoutthischapter,weassumethatanytwosensornodescandirectlycommunicateviabi-directionalwirelesslinksiftheirEuclideandistanceisnotgreaterthanrc,thecommunicationrange;andapositionintheplanecanbeperfectlymonitored(orcovered)byasensornodeiftheirEuclideandistanceisnotgreaterthanrs,thesensingrange.Similarto[ 10 93 159 ],wealsoassumethatsensornodesarehomogeneousinthesensethatrcandrsarethesameforallnodes,andkeepconstantduringeachnode'slifetime.Insteadofconsideringallthepossiblecombinationsofrcandrs,wefocusonthecaseofrc=2rsinthischapter.Therearetworeasonsfordoingso.First,aspointedoutin[ 174 ],thespecicationofrc2rsholdsformostcommerciallyavailablesensorssuchasBerkeleyMotesandPyroelectricinfraredsensors.Second,asshowninSection 2.3.6 ,forarbitraryspatialdistributionsofsensornodes,rc2rsisthesufcientandnecessaryconditionfortheexistenceoflocalboundarynodedetectionalgorithms1.Therefore,wesetrc=2rstoreducecommunicationenergyconsumptionandinterference.However,itshouldbenotedthatouralgorithmsarestillapplicabletothescenariosofrc>2rswithoutanychanges.WealsopresentaschemecalledSABtodealwiththecaseofrc<2rs(cf.Section 2.5.2.2 ).Forsimplicity,weassumethattheROIisa2-Dsquareplanareldhereafter.Ourresults,however,canbeeasilyextendedto2-Dor3-DROIsofarbitraryshapes.Forl>0,letAldenotethesquareROIofsidelengthlcenteredattheorigin,i.e.,Al=[)]TJ /F3 11.955 Tf 9.3 0 Td[(l=2,l=2]2,and@AlbetheborderofAl.Weexaminealarge-scaleWSNconsistingof 1TheformaldenitionofboundarynodesandlocalalgorithmswillbegiveninSections 2.3.1 and 2.3.6 ,respectively. 26

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hundredsoreventhousandsofstationarysensornodes2,anddenotethesensornodesdeployedintheROItobeV=fs1,,si,,sng(si2Al,for1in,i2N),wheresirepresentsthepositionofnodeiandnisthetotalnumberofsensornodes(ornetworksize).Ingeneral,noassumptionshouldbemadeaboutthedistributionofthesensornodesintheenvironment.Ourschemesaredesignedtoworkcorrectlyunderarbitrarynodedistributions.However,intheperformanceevaluationoftheschemesproposedinthischapter,weutilizehomogeneousSpatialPoissonPointProcess(SPPP)asthenodedistributionmodeltofacilitatethetheoreticalanalysisandsimulations.Itiswellknownthatthismodelisagoodapproximationofthedistributionofsensornodesmassivelyorrandomlydeployed(e.g,viaaerialscatteringorartillerylaunching)andcanbeeasilyextendedtocharacterizetheprocessthatnodesfaildynamically. 2.2.2DesignGoalsInthischapter,weintendtodesignacoverageinferenceprotocolwhichcanprovidetheBSanaccurateandtimelymeasurementofthecurrentconnectedcoverageinanenergy-efcientway.Specically,ourdesigngoalsinclude:Effectivenessandrobustness:TheBSshouldbeabletohaveatimelyandaccurateviewoftheconnectedcoverage,regardlessofarbitrarynetworktopologies,locationerrorsofsensornodesanderror-pronewirelesschannelssuchthatitcaninstructnecessaryandquickactions,e.g.,addingnewsensornodestoenlargetheconnectedcoverage. 2StationarynodesheredonotimplythatthetopologyoftheWSNisstatic.Instead,theWSNmayhavehighlydynamictopologychangesduetonodefailures,newnodeadditionsornodesswitchingtheirstatesbetweenactiveandsleepingmodestosaveenergy.OneadvantageofourschemesliesintheefciencytohandletopologychangesinWSNs(cf.Section 2.3.5 ). 27

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Trulylocalizedanddistributedproperties:Ascomparedtopreviousapproaches,thecoverageinferenceprotocolisintendedtobeatrulylocalizedanddistributedsolution,inwhicheachsensornodecanself-determinewhetheritisonthearea-coverageboundarybyafewsimplecomputationsoninformationonlyfromone-hopneighbors.Thesenicepropertieswillenabletheprotocoltohavelowcomputationalandcommunicationoverhead,highenergyefciency,andexcellentnetworkscalability.Universalapplicability:AlthoughWSNsareoftensaidtobehighlyapplication-dependent[ 4 79 ],thecoverageinferenceprotocolisdesignedtoworkwitharbitraryapplicationsandnetworktopologiesandtobeindependentofalltheothercomponentsofthenetworkprotocolstack.Versatility:Forresource-constrainedsensornetworks,itishighlydesirablethatsomebasicprotocoloperationsforimplementingonefunctionalitycanbereusedinprovidingothernecessaryfunctionalities.Otherwisetheprotocolstackofsensornodeswillbetoocomplicatedtohavehighoperationalefciency[ 79 ].Thedesignofcoverageinferenceprotocolwilltakethisrequirementintoconsiderationsothatmanyofitsbasicoperationscanbehighlyreusedinrealizingimportantfunctionalitiesotherthannetworkhealthdiagnosis,aswewillshowsoon. 2.3BOND:BoundaryNodeDetectionSchemeThissectionpresentsBONDwhichenableseachnodetolocallyself-detectwhetheritisaboundarynode.Webeginwithseveralimportantdenitions,followedbytheillustrationofBOND. 2.3.1BoundaryNodeandItsDetectionAlgorithmWesaythatnodessiandsj(i6=jandsi,sj2V)areneighborsorthereexistsadirectwirelesslinkbetweenthemiftheEuclideandistancebetweenthemisnolargerthanrc,i.e.,ksi)]TJ /F3 11.955 Tf 11.95 0 Td[(sjkrc.WealsodenotebyNeig(si)theneighborsofnodesi(notincludingsi).Inaddition,twonodessiandsjaresaidtobeconnectedifthereisatleastonepathconsistingofdirectwirelesslinksbetweenthem,similarlyasetofnodesis 28

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Figure2-1. AnexemplaryWSNs.Therearethreesensorclusters:Clust(s1),Clust(s2),andClust(s3),twostaticBSs:BS1andBS2andonemobileBS:MS.Shadedarea,soliddotsandopendotsrepresentthecoverageofsensors,boundarynodesandinteriornodes,respectively.TheboundaryofCover(s2)isdepictedbyredcurves. calledconnectedifatleastonepathexistsbetweeneachpairofnodesintheset.ThefundamentalconnectedunitofWSNsiscalledacluster: Denition2.1. [cluster]Aconnectedsetofnodesissaidtobeaclusteriftheinclusionofanyothernodenotinthissetwillbreaktheconnectednessproperty.WewriteClust(si)fortheclustercontainingnodesi.Basedonthesensingmodel,thesensingdisk(orcoverage)ofnodesicanbegivenby Diski=Disk(si,rs)=fu2R2:ku)]TJ /F3 11.955 Tf 11.95 0 Td[(sikrsg.(2)Specically,let0indicatetheorigin,wehaveDisk0=Disk(0,rs).Thenthecoveragecorrespondingtoaclustercanbedenedasfollows: Denition2.2. [coverageofacluster]GivenClust(si),werefertothesetofallpointsinthemonitoredeldthatarewithinradiusrsfromanynodeofClust(si)asthesetcoveredbyclusterClust(si).DenotingthissetbyCover(si),wehave Cover(si)=0@[u2Clust(si)(u+Disk0)1A\Al.(2)Obviously,Cover(si)isuniquelydeterminedbyitsboundary@Cover(si). 29

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Denition2.3. [boundaryandinteriornode]WedeneboundarynodesofClust(si)asthosewhoseminimumdistancesto@Cover(si)areequaltors,i.e., BN(si)=fu2Clust(si):minku)]TJ /F3 11.955 Tf 11.96 0 Td[(vk=rsforv2@Cover(si)g.(2)Accordingly,wecallalltheothernodesinClust(si)asinteriornodes,i.e., IN(si)=fu2Clust(si):u=2BN(si)g.(2)Theminimuminformationneededtodescribe@Cover(si)isrcandBN(si).WedenotethepositionofthebasestationasBS,thentheconnectedcoverageoftheBSis3 Cover(BS)=S1inCover(si):BST)]TJ /F3 11.955 Tf 5.47 -9.68 Td[(Clust(si)Disk(0,rc)6=;.(2)Obviously,theproblemofndingtheboundaryofconnectedcoverage,i.e.,@Cover(BS),isequivalenttodetectingtheboundarynodesofclusterswithconnectionstotheBS.Basedonthisobservation,itispossibletodesignadistributedCIPifwecanrstndalocalizedwaytodetectboundarynodes.NotethatourdenitionofboundarynodesisuniquelybasedontheclustertheybelongtoandisunrelatedtothepositionoftheBS.Inaddition,ourdenitionofconnectedcoverageisapplicabletoWSNswithmultipleormobileBSs.Forexample,intheWSNgiveninFigure 2-1 wheretherearethreeBSs(twostaticandonemobile),fromEq.( 2 )wecandirectlyobtaintheconnectedcoverageoftheWSNattimeinstancest1andt2asCover(s1)SCover(s2)SCover(s3)andCover(s2)SCover(s3),respectively.Sinceforthemultiple-BScases,theconnectedcoverageoftheWSNisjust 3NotethatAB=fu+v:u2A,v2BgforA,BR2. 30

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Figure2-2. IllustrationoftheLVP-basedboundarynodedetectionalgorithm. theunionoftheconnectedcoverageregardingeachindividualBS,hereafterwefocusonthesingle-BScasefortheeaseofpresentation. 2.3.2LocalizedVoronoiPolygonsOurBONDschemeisbasedontwonovelgeometricconceptscalledLocalizedVoronoiPolygon(LVP)andTentativeLVP(TLVP)whicharenontrivialgeneralizationofVoronoiPolygons(VPs)[ 123 ]fromcomputationalgeometry.WemustpointoutthatasimilarconceptcalledLocalizedVoronoiDiagrams(LVDs)isintroducedasthedualofLocalizedDelaunayTriangulations(LDTs)intheliterature[ 78 98 ].TheedgecomplexityofLDTisanalyzedin[ 78 ]anditsapplicationsintopologycontrolandroutingforwirelessnetworksarediscussedin[ 98 ].However,thereisnoindicationonhowtorelatethisconcepttothecoverageproblemsinWSNs.Moreover,unlikeourworkthereisnodescriptiononhowtoefcientlyconstructLVDsgivenin[ 78 98 ].FurthermoretheideaofusingTLVPtoreducetheoverheadofthedetectionalgorithminthischapteriscompletelynew.Finallyandmostimportant,ourschemeBONDonlyusesthelocalinformationtodetectboundaryinsteadofglobalinformationcommonlyusedineitherVPorDT.WerstdeneVPs,LVPsandTLVPsintermsofhalfplanes.Fortwodistinctpointssi,sj2V,thedominanceregionofsioversjisdenedasthesetofpointswhichareatleastasclosetosiastosj,i.e., Dom(si,sj)=v2R2:kv)]TJ /F3 11.955 Tf 11.95 0 Td[(sikkv)]TJ /F3 11.955 Tf 11.95 0 Td[(sjk.(2) 31

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Obviously,Dom(si,sj)isahalfplaneboundedbytheperpendicularbisectorofsiandsj,whichseparatesallpointsintheplaneclosertosithanthoseclosertosj. Denition2.4. [VP,LVPandTLVP]TheVPassociatedwithsidenotedbyVor(si),isthesubsetoftheplanethatliesinallthedominanceregionsofsioverotherpointsinV,namely, Vor(si)=\sj2VfsigDom(si,sj).(2)Inthesameway,theLVPdenotedbyLVor(si),andtheTLVPdenotedbyTLVor(si),associatedwithsiaredenedas: LVor(si)=\sj2Neig(si)Dom(si,sj);(2) TLVor(si)=\sj2SubNeig(si)Dom(si,sj),(2)whereSubNeig(si)isapropersubsetofNeig(si),i.e.,SubNeig(si)Neig(si).ThecollectionofLVPsgivenby LVor(V)=fLVor(si):si2Vg(2)iscalledthelocalizedVoronoidiagram(LVD)generatedbythenodesetV.TheboundaryofLVor(si),i.e.,@LVor(si),mayconsistoflinesegments,halflines,orinnitelines,whichareallcalledlocalVoronoiedges. Lemma1. PropertiesofVPs,LVPsandTLVPs:(i)LVor(si),TLVor(si)andVor(si)areconvexsets;(ii)Vor(si)LVor(si)TLVor(si);(iii)PlaneR2iscompletelycoveredbyLVor(V). Proof. (i)Sinceahalfplaneisaconvexsetandtheintersectionofconvexsetsisaconvexset,aLVP(oraTLVP)aswellasaVPisaconvexset.(ii)FromEqs.( 2 ),( 2 )and( 2 )wehaveVor(si)=LVor(si)\\sj2V,sj=2Neig(sj)Dom(si,sj), 32

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LVor(si)=TLVor(si)TTsj2Neig,sj=2SubNeig(sj)Dom(si,sj),whichdirectlyleadstoLemma 1 (ii).(iii)Itiswellknownincomputationalgeometrythat [si2VVor(si)=R2.(2)(cf.[ 123 ,PropertyV1,pp.77]).Combining( 2 )withLemma 1 (ii),wecandirectlyobtainLemma 1 (iii). Therefore,thesetLVor(V)\AcanfullycoveranarbitrarysubsetAR2.NotethatthisresultcanbeeasilyextendedtoanyclusterClust(si)inV,thatis, [sj2Clust(si)LVor(sj)=R2.(2) 2.3.3LVP-BasedBoundaryNodeDetectionInthissubsection,wepresentourBONDschemeforeachnodetodetectwhetheritisaboundarynodebasedonitsownLVPorTLVPbytakingnodesiasanexample. 2.3.3.1InputOurBONDisadistributedschemeinthatweonlyneedpositionsofnodesi'sneighborsastheinputofouralgorithm.WetemporallyassumethatthereisnolocationerrorandwillrelaxthisassumptioninSection 2.6.1 .WeneedtoconsidertwocasesbasedonwhethertheinformationabouttheborderofAl,i.e.,@Al,isavailable.Intherstcasewhen@Alisunavailableatnodesi,ourdetectionschemeisbasedontheconstructionofLVor(si)(orTLVor(si));inthesecondcasewhen@Alisavailable,weneedtoexploitthisinformationbycalculatingLVor(si)\Al(orTLVor(si)\Al).ItcanbeshownthatLVor(si)\Almustbeaniteconvexpolygon.Thus,thesecondcasecanbetransformedintotherstcasebyintroducingdummynodesintoNeig(si).SeeFigure 2-2 (a)forexample,inwhichfourdummynodes,d1throughd4,areintroducedsuchthat 33

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perpendicularbisectorsbetweensiandthedummynodesgeneratethefourborderedgesofROI.ThenwecancalculateLVor(si)\AlbyfollowingthesameprocedureforcalculatingLVor(si).Therefore,wewillonlydiscusstherstcaseinwhatfollows.Wenoticethatdummynodescannotbedirectlyappliedtothegeneralizedcases,i.e.,theborderofAlconsistingofcurves.Howeverinthesecases,thisjustmeansthattheinformationofborderofAl'sbordercannotbeefcientlyexploited,andthecorrectnessofourschemeisnotaffected.TherealsoexisttwoeasywaystoremedyourBONDhere.First,ingeneralacurvecanbeapproximatedwithstraightlinesegmentsandthustheBONDisstillapplicable.Second,insteadofcheckingwhethertheverticesofLVor(si)\AlarecoveredbyDisk(si,rs)whenAlisapolygon,westillcancorrectlydetectboundarynodesbycheckingeverypointon@(LVor(si)\Al)whenAlisnotapolygon. 2.3.3.2AlgorithmOurgoalistoconstructtheLVor(si)(orTLVor(si))whichissufcientfortheboundarynodedetectionwiththeminimalrequirementontheinformationaboutsi'sneighbors.WerstdivideDisk(si,rc)intofour4quadrants.ThenweconstructtheTLVPofsibyusingthenearestneighbors(solidnodesinFigure 2-2 (b))ineachofthefourquadrants.Withoutloseofgenerality,wedenotethesefournearestneighborsass1,s2,s3,ands4.TherstTLVPiscalculatedbyTLVor(si) \4j=1Dom(si,sj).IfallverticesoftheTLVPiscoveredbyDisk(si,rs),theprocedurestopsandthisTLVPissaved.Otherwise,weneedtondnewneighborswhicharethenearesttotheuncoveredverticesoftheTLVP(cf.Figure 2-2 (c)),addthoseneighborstoSubNeig(si), 4Othervalueswillalsoworkwell. 34

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andcalculatetheTLVPagain:TLVor(si) TLVor(si)TTsj2SubNeig(si),j6=1,2,3,4Dom(si,sj).ThenewverticesofthenewTLVPwillbecheckedtoseewhethertheyarecoveredbyDisk(si,rs).ThisprocedurecontinuesuntilalltheverticesoftheTLVParecoveredbyDisk(si,rs)ortheLVPofsiiscalculatedandsaved.Notethatwhen@Alisunavailable,LVor(si)maybeinnite,whichmeansthatitispossiblethatwecannotndanynodesinoneormorequadrantsintherststep.SeeFigure 2-2 (d)foranexample.Ifaquadrantcontainsnoneighbors,wedenetwosectorsofangle45whicharedirectlyadjacenttothequadrantastheassistantarea,andaddthenodesinthisareatoSubNeig(si)rst.IfallthenodesintheassistantareacannotmakeTLVPnite,wecanconcludethatLVPmustbeinnitewithoutneedtodofurthercalculation. 2.3.3.3OutputIfLVor(si)isinnite,simustbeaboundarynode.IfLVor(si)(orthenalTLVor(si))isnitewithalltheverticesarecoveredbysi,thensi2IN(si).Otherwise,si2BN(si). 2.3.4AlgorithmValidationIntheVD,theVPsofdifferentnodesaremutuallyexclusive,butintheLVD,theLVPsofdifferentnodesmayoverlap.ThisdifferencemakesthevalidationofouralgorithmtotallydifferentfromthatofexistingVP-basedones. Theorem2.1. Ifthereisapointv2LVor(si)whichisnotcoveredbysi,i.e.,v=2Disk(si,rs),theremustexistapointh2LVor(si)thatisnotcoveredbyanynode,andsimustbeanarea-coverageboundarynode. Proof. Withoutlossofgenerality,weassumethatthenodenearesttosiandoutsideDisk(si,rc)issmandksi)]TJ /F3 11.955 Tf 11.96 0 Td[(smk=rc+for>0.Lets`mbethepointon sivsatisfyingksis`mk=ksismk,andhbeanotherpointon sivsuchthatksihk=rs+/2(Figure 2-3 ). 35

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Bythetriangularinequality,wehaveksmhk+ksihkksismk=ksis`mk=ksihk+khs`mk.Therefore,ksmhkkhs`mk=ksis`mk)-26(ksihk=rs+/2,whichmeansthatsmcannotcoverhandnordoesanyothernodeinDisk(si,rc)C.Thereasonisthat,sinceksislk>ksismkholdsforanynodesl2Disk(si,rs)Candsl6=sm,wehaveks`lhk>ks`mhkwherepoints`lisontheline sivandksis`lk=ksislk.Therefore,kslhkks`lhk>ks`mhk=rs+/2.Sincev2LVor(si),basedontheconvexityofLVor(si)wehave siv2LVor(si).Therefore,h2LVor(si),whichimpliesforanynodesj2Disk(si,rc)andsi6=sj,wehaveksjhkksihk>rs,i.e.,nonodesinDisk(si,rc)cancoverh.Consequently,wecanconcludethatnonodeintheplanecancoverhbecauseDisk(si,rc)[Disk(si,rc)C=R2.Notethatfromtheaboveproofprocess,wecanseethathcanbearbitraryclosetov`,theintersectionofcircle@Disk(si,rs)and siv.Therefore,siisaboundarynode. Theorem2.2. Ifthereisapointv2Alnotcoveredbyanysensornode,foreveryclusterClust(si)theremustexistatleastonesensorsj2Clust(si)whoseLVor(sj)isnotcompletelycoveredbyDisk(sj,rs). Proof. AccordingtoLemma 1 (iii)or( 2 ),wehave [sj2Clust(si)(LVor(sj)\Al)=Al(2)Therefore,foranyv2Al,itmustlieinatleastoneLVor(sj)\Alforsj2Clust(si). Theorems 2.1 and 2.2 provethatthesufcientandnecessaryconditionforClust(si)tocompletelycoverAlisthatLVor(sj)\Aliscompletelycoveredbysjforallsj2Clust(si).ThefollowingtheoremshowsthatwhenLVor(si)orLVor(si)\Alisnite,thecoverageofverticesofLVor(si)bysiisequivalenttothecoverageofthewholeLVor(si)bysi,whichguaranteesthecorrectnessofourLVP-basedalgorithm. Theorem2.3. LVor(si)isfullycoveredbysiifandonlyifLVor(si)isniteandalltheverticesarecoveredbysi. 36

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Figure2-3. IllustrationoftheproofofTheorem 2.1 Proof. LetVe(si)bethesetofverticesofLVor(si).Obviously,whenLVor(si)iscompletelycoveredbysi,i.e.,LVor(si)Disk(si,rs),wehavev2Disk(si,rs)forallv2Ve(si)andLVor(si)isnite.Sincemaxu2LVor(si)fksi)]TJ /F3 11.955 Tf 11.95 0 Td[(ukgmaxv2Ve(si)fksi)]TJ /F3 11.955 Tf 11.95 0 Td[(vkg,whenv2Disk(si,rs)forallv2Ve(si),wehaveu2Disk(si,rs)forallu2LVor(si). 2.3.5DiscussionsonBONDLowOverhead.Ithasbeenshownin[ 123 ]thatingeneralVPscannotbecomputedlocally.Therefore,thetraditionalVP-basedschemes[ 40 53 157 ]arenotdistributedandareveryexpensiveintermsofcommunicationoverhead.OurBONDschemeisatrulylocalizedpolygon-basedsolutionbecausecomputingLVor(si)(orTLVor(si))onlyneedsone-hopinformation(thiscanbedirectlyobtainedfromtheEqs. 2 and 2 ).Assumingthatthenumberofneighborsisk,eachnodecancomputeitsownLVor(si)withcomplexitysmallerthanO(k).Inaddition,thecomputationoftheLVor(si)onlyinvolvessomesimpleoperationsonpolygonswhichcanbeefcientlyimplemented(e.g.,PolyBooleanlibrary[ 97 ]).WefurthersimplifythedetectionprocessbyconstructingTLVPsrst.ForadenselydeployedWSN,wehaveLVor(si)orTLVor(si)!Vor(si),anditiswellknownincomputationalgeometrythat 37

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underthehomogeneousspatialPoissonpointprocess,theaveragenumberofverticesofVor(si)is6[ 123 ].Therefore,whenthenodedensityishigh,BONDonaverageonlyneeds4to6nearestneighbors'informationtosuccessfullydetecttheboundarynodes.Moreover,whenaneighbornodedies,BONDneedsdonothingunlessthedeadnodeisusedtoconstructthenalTLVor(si)orLVor(si)inthelastturnofLVPorTLVPconstruction.Thisuniquepropertywillgreatlysimplifytheupdateofdetectionresultsandsavepreciousenergyofeachsensornode.Alltheseadvantagescannotbeachievedbyotherlocalizedboundarynodedetectionschemesintheliterature,suchastheperimeter-coveragecheckingapproach[ 69 ]andthecrossing-coveragecheckingapproach[ 62 174 ].WerefertoSection 2.5.1 foradetailedcomparison.OtherApplications.WeareawareofthefollowingapplicationsofBONDoritsbasicoperations,whicharebynomeansacompletelist. Topologycontrolandrouting.Ithasbeenshownin[ 78 98 ]thatthedualofLVP,calledLDT,canbeusedtodesigndistributedtopologycontrolandroutingprotocolsforwirelessadhocnetworkswithenergyefciencyandtheguaranteeofthedelivery.Fromthepropertyofduality,wecandirectlyobtainLVPsifLDTsaredetermined,orviseversa. Spatialaggregation.IndistributeddataprocessingforWSNs,toreducethesamplingerrorsintheaggregatedspatialdata,itisproposedin[ 138 ]torstcalculatetheVPofeachsensor.Asmentionedbefore,sincetheVPcannotbecomputedlocally,theLVPcanbeusedasagoodapproximationoftheVPinspatialaggregation. Coverage-preservingnodesleepingscheduling.Sincesensornodesareusuallydeployedwithredundancy,itispossibletoprolongthenetworklifetimewhilepreservingtheconnectedcoveragebyschedulingsomenodesintothesleepingstate[ 153 ].EachnodecanlocallydecidewhetheritsownLVPiscoveredbyitsneighbors.Ifthisisthecase,anodedeclaresitselfeligibleforsleeping,announcesthisfacttoitsneighbors,andthengoestosleep.Toavoidtheformingofcoverageholecausedbyeligiblenodesswitchingintosleepsimultaneously,arandomlydelayedannouncementschemeusingcommonrandombackoffapproachesisproposedin[ 153 ]. 38

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Figure2-4. Non-localityoftheboundarynodedetectionwhenrc<2rs. 2.3.6LocalityofBoundaryNodeDetectionInthissubsectionweinvestigatefurthertoshowthatitisimpossibletondlocalalgorithmsforboundarynodedetectionwitharbitrarynodedistributionswhenrc=rs<2.Werstdenewhatwemeanbylocalalgorithms.Thisdenitionisbasedonamodelproposedin[ 128 ]. Denition2.5. [LocalAlgorithm]Assumethateachcomputationsteptakesoneunitoftimeandsodoeseverymessagetogetfromonenodetoitsdirectlyconnectedneighbors.Withthismodel,analgorithmiscalledlocalifitscomputationtimeisO(1),intermsofthenumberofnodesninthesystem.OurBONDshowsthatwhenrc=2rs,sensorscanlocallydetermineifitisaboundarynode.Whenrc>2rs,sincethenodewillhavemoreinformationaboutothernodesarounditself,itcanstilllocallydetectwhetheritisaboundarynode.However,inthecaseofrc<2rs,individualnodescanneitherlocallysayyesnornotothequestionofwhetheragivennodeisaboundarynode.Toseethis,considersensorsdeployedasinFigure 2-4 .Obviously,nodesiisaninteriornode.However,toconrmthis,itneedstoknowtheexistenceofnodess1tos5withthehelpofsomerelaynodes(greennodes).InFigure 2-4 (a),nodes4isalready5hopsawayfromnodesi.Infactthedistancebetweenthesetwonodescanbearbitrarylong,whichisshowninFigure 2-4 (b).Therefore,forarbitrarynodedistributions,itisimpossibletondalocalized 39

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Figure2-5. BasicoperationsoftheBOND-basedCIP.(a)EachsensorexecutesBONDindividually.(b)Boundarynodes(blackdotsin(a))reportthemselvestothebasestation.(c)TheBSreconstructsthecoverageboundary.Notethattheshadedareain(a)representsthecoverageofsensors,andthattheshadedareaattheleftbottomcornerin(a)islostin(c)becauseitisnottheconnectedcoverage. boundarynodedetectionalgorithmthatalwaysworks.In[ 10 ],theauthorsconsideredgeneralvaluesofrc=rswithregulardeploymentpatternssuchasthehexagon,squaregrid,rhombus,andequilateraltriangle.Unlike[ 10 ],inthischapter,weprefertomakethestrictassumptiononthevalueofrc=rsratherthanonthenodedistributionpattern.Thereasonisthateveninsomescenarios,sensornodesareoriginallydistributedwithregularpatterns(e.g.,hexagon,squaregrid,etc.),duetonodefailures,nodemovementscausedbyanimalsorwinds,etc.,thetopologyoftheWSNwillbecomeirregularsoonerorlater.Incontrast,eveninthecaseofrc=rs<2,wecanstillassumeasmallervalueofrswherebytogetaconservativeinferenceofthecoverage,whichisdesirableforsomesecurity-criticalapplications. 2.4CIP:CoverageInferenceProtocolInthissection,wedescribehowtouseBONDtobuildapracticalcoverageinferenceprotocol(CIP). 40

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Ourdesignphilosophyisthat,sincetheminimuminformationrequiredtodescribethecoverageisthepositionsofboundarynodes(cf.Section 2.3.1 ),wejustneedtodetectboundarynodes.Inotherwords,ourschemecanensurethat,fortheBStoreconstructthecoverageimagewithoutanydistortion,theinformationtransmittedfromsensorstotheBSisminimized.AlsonotethatourBONDonlyinvolveslocalmessageexchanges.Inalarge-scaleWSN,theoverheadfromlocalbroadcastisverysmallascomparedtothatfromtheend-to-endcommunicationsfromsensornodestotheBS.Therefore,ourapproachcansavethepreciousenergyofsensornodes.Figure 2-5 illustratesthebasicoperationsofourBOND-basedCIP,whichconsistofthefollowingthreesteps: 2.4.1NeighborhoodMonitoringandSelf-DetectionAfterthedeploymentoftheWSN,weassumelocalizationtechniquesareavailableforsensornodestodecidetheirpositions.Eachnodethencollectsthepositioninformationofitsneighborsbybroadcastingitsownpositions,andexecutesBONDtodetectwhetheritisaboundarynode.Ifso,itwillreportitspositiontotheBS.WerefertotheneighborsusedtoconstructtheLVPorTLVPinthelastrunofBONDasitsconsultingneighbors.Inourprotocol,bothinteriorandboundarynodesarerequiredtobroadcastanExistenceUpdatingPacket(EUP)totheirneighborsforarandomperiodoftimeexponentiallydistributedwithrateTEUP.Inaddition,eachinteriornode,saysi,maintainsatimerC0(j)ofexpiryvaluemuchlargerthanTEUPforeachofitsnon-consultingneighbors,saysj.Ifsidoesnotoverhearanypacket(eitheranEUPordatapacket)fromsjbeforeC0(j)expires,itwilltreatsjasadeadneighbor,whichcanbecomealiveifsioverhearsanypacketfromitlater.Nodesialsomaintainstwotimersforeachofitsconsultingneighbors,saysk:theneighbor-monitoringtimerC1(k)andtheneighborquerytimerC2(k).IfsidoesnotoverhearanypacketfromskbeforeC1(k)expires,itunicastsaNeighborQueryPacket(NQP)toskandstartsC2(k).Ifstillalive,skis 41

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requiredtosendbackanEUPimmediatelyandwaitforanACKfromsi.IfnodesistilldoesnotoverhearanypacketfromskbeforeC2(k)expires,siwilltreatskasadeadneighborandre-executeBONDwithaliveneighborsasinput.Ingeneral,theexpiryvaluesofC0(j)andC1(k)shouldbeinthesameorderofTEUP,inordertoguaranteethatwithhighprobability,eachnodewillreceiveEUPsfromallalivenodesinitsneighborhood.TheexpiryvaluesofC2(k)shouldbemuchsmallerthanTEUP,becausewerequirethatthenodewhichreceivestheNQPneedstosendbackanEUPimmediately.Unlikepreviousneighbor-monitoringschemesemployingasingletimer[ 137 ]ortreatingneighborsasthesame[ 68 ],ourschemesetsdifferenttimersfornon-consultingandconsultingneighbors.ThemajorreasonfordoingsoisthatdatapacketsandEUP-likebroadcastpacketsaresubjecttolossduetowirelesstransmissionerrorsorcollisions.Asaresult,anodemayfalselyidentifyanaliveneighborasadeadone.Obviously,fornon-consultingneighbors,wecandecreasethefalsepositiveratebysettingalargertimervalue.However,usingalargertimervalueforconsultingneighborswillincreasetheresponsedelay,i.e.,thedelayfromwhencoverageholesemergetowhentheyaredetectedbyboundarynodes.Therefore,weusetwotimersforconsultingneighborstoensurebothashorterresponsedelayandalowerfalsepositiverate:althoughtheexpiryvalueofC1(k)issmall,weactivelyquerythequestionableneighborbeforewetreatitasthedeadneighbor,whichmaysignicantlyincreasetheaccuracyofourscheme.Asmentionedbefore,ifthenodedistributionfollowsSPPP,eachnodeonlyhas4to6consultingneighborsonaverage,whichmeansthehighfeasibilityofourtwo-timerscheme.Therefore,byadoptingone-timer(C0)schemefornon-consultingneighborsandtwo-timer(C1andC2)schemeforconsultingneighbors,ourdesignachievesabetterbalanceamongaccuracy,delayandcommunicationoverhead.NotethatthisbenetstemsfromthefactthatourBONDdivideseachnode'sneighborsintotwocategories.Forotherboundary-node-basedapproachesintheliterature(cf. 42

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Section 2.5.1 )wherethereisnodivisioninneighbors,eitherone-timerortwo-timerschemeshouldbeadoptedforallneighborsandthebalancebetweencommunicationoverheadandtheaccuracycannotbehandledinthisway.Notethatthecommunicationoverheadofourneighbor-monitoringschemecanbefurtherreduced.TheEUPpacketscanpiggybackontoregularlocalbroadcastpacketsusedtolearnlinkconditions,maintaintheroutinginformation,andfacilitateothernetworkoperations.Inaddition,inthepresenceofdatatrafc,anypacketoverheardbyanodeshouldberegardedasanEUPpacket,andanydatapacketsentfromanodecancancelthenextEUPpacketitshouldbroadcast. 2.4.2Self-ReportingofBoundaryNodesWheneveridentifyingitselfasaboundarynode,asensornodeshouldsenditspositioninformationtotheBSwhichcanreconstructtheimageofthecoveragebasedonallthereceivedpositioninformationofboundarynodes. 2.4.3ExplicitACKsfromtheBSSincethepacketlossratioduetocollisionsornoiseisprettyhighintheWSN[ 79 ],boundarynodesneedsomemechanismstoensurethattheirreportshavebeenreceivedbytheBS.Otherwise,theyhavetorepeatedlyresendtheirreports,whichcausesenergywaste.Theissueofreliablesensor-to-BScommunication,thusfar,hasnotbeenaddressedthoroughlyinWSNresearchcommunity.TheworkonreliablecommunicationinWSNsrstappearsin[ 155 ],andthenin[ 149 ].However,noguaranteedreliabilitysemanticsareprovidedinthesework.In[ 136 ],theauthorsrstlyproposethenotioncalledevent-to-sinkreliability,andstudythereliablecommunicationfromthisperspective.Theirsolutionismainlybasedonadjustingthereportingfrequencyofsourcenodes,andisapplicableformonitoringacontinuouslychangingeventorreportingahugeamountofdata.Forourcase,eachsensornodewillbeontheboundaryornotforarelativelylongperiod,andeachboundarynodeonlyneedstoreportitslocationinformationtotheBS.Therefore,theschemeproposedin[ 136 ]cannot 43

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beappliedhere.In[ 127 ],theauthorsstudytheBS-to-sinksreliability,andarguethattherequirementandimplementationofreliabilityinaWSNisrmlydependentuponthespecicapplication,andthereisnoone-for-allsolution.Forourproblemseating,anintuitivesolutionistorequiretheBStosendindividualACKtoeachboundarynodefromwherethereporthasbeenreceivedattheBS.Inwhatfollows,wewillshowthatthisintuitivesolutioncanbeimprovedbyintroducingtheBloomlter[ 15 ].Forthesakeofclarity,westartwithdescriptionofACKformatwiththeBloomlter.Thenweanalyzeitsperformancequantitatively.ThebasicideahereistouseBloomlterstodesignanenergy-efcientapproachfortheBStobroadcastonlyoneACKtoacknowledgemultipleboundarynodes.Lets1,s2,,sabetheaboundarynodeswhichtheBSwantstoacknowledgeexplicitly.Leth1,h2,,hbbethebhashfunctionsoftheBloomlter,eachwithrangef1,2,,lg.Letack(t)=(b1,b2,,bl)beabitvectoroflengthl.ack(t)isthet-thBloomlterusedtoindicateboundarynodestheBSwantstoacknowledge,anditisinitiallysetto(0,0,,0).For8i,1iband8j,1ja,theBScomputeshij=hi(IDsj)whereIDsjindicatestheuniqueIDofnodesj,andsetsbhij=1.TheBScanthenusetheBloomlterack(t)asACKandsenditbackusingefcientbroadcastorgeographicmulticastprotocolsforWSNs.Whenaboundarynodesjreceivestheack(t),itperformsamembershiptest:itcomputeshi(IDsj)for8i,1ib;ifallofthesepositionsare1intheack(t),thenboundarynodesjknowsitsreporthasbeenreceivedandacknowledgedbytheBS.NotethattheBloomltermayinduceasmallnumberoffalsepositives,i.e.,afewun-acknowledgedboundarynodesmaypassthemembershiptestandthereforebelievethattheirreportshavebeenreceivedbytheBS.Ontheotherhand,Bloomltersensurethattherearenofalsenegatives,i.e.,allacknowledgednodesareguaranteedtopassthemembershiptest.Inpracticewecantuneourbandlparameterstoenabletradeoffbetweencommunicationandcomputationaloverheadandfalsepositiverate. 44

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NowweanalyzethebenetofutilizingACKswithBloomlters.Firstofall,weneedquantifythecommunicationoverheadofindividual-ACKscheme.Supposetherearensensornodesinthenetwork,andthelengthofeachnodeIDislognbits.IfthereareaboundarynodeswhichtheBSwantstoacknowledgeexplicitly,theBSneedsendoutaACKpackets,andeachpacketwiththesizeofO(logn)bits.ThisisbecausetheheaderoftheACKpacketshouldincludethedestinationnodeIDwithlognbits.Next,weconsiderthemulti-hopcommunicationschemethatcanbeusedtosupporttheBS-to-sensorcommunication.Notethateachsensornodeonlyhasanitebuffer,andthebuffersizeisrelativelysmallcomparedtonforalarge-scaleWSN.Therefore,itisalmostimpossibletoprovideunicastcommunicationfromtheBStoindividualsensornode.NotethattosupporttheunicastcommunicationfromtheindividualsensortotheBSisstillpossible(cf.Section 2.5.2.2 ).ThisiscausedbytheasymmetryoftrafcpatterninWSNs[ 79 ].Forthesensor-to-BSunicastcommunication,wecanconstructaroutingtreerootedattheBS(refertoFigure 2-11 foranexample),andeachnodeonlyneedstorelaythepacketfromitschildnodestoitsparentnodeinthisroutingtree.Tomaintainthisroutingtree,eachnodeonlyneedstoaddoneentryintoitsroutingtablewiththedestinationastheBSandthenext-hopasitsone-hopparentnode.FortheBS-to-sensorunicastcommunication,thesituationwillbetotallydifferent.Foreverychildnodeinthesubtreerootedataparticularnode,thisnodeneedstoaddoneentryinitsroutingtableforthatchildnode.ForthenodeneartheBS,itmayneedO(n)entriesintheroutingtable,whichistoolargeforthebuffer-constraintsensors.So,inpractice,therearenoefcientschemetosupportunicastdownstreamcommunication(fromtheBStosensors)[ 79 ].Especiallyforthesmallsizepacket,toestablishatentativeroutebybroadcastaroutediscoverypacketisinvaluable.Therefore,evenfortheBS-to-sensorunicastcommunication,whenthepacketsizeissmall,westilluseglobalbroadcastfromtheBS. 45

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Inindividual-ACKscheme,theBSneedstoperformabroadcasts,eachwithpacketsizeO(logn)bits.ForBloom-lter-basedscheme,theBSneedstoperformonebroadcastwithpacketsizeO(l)bits.ForthestandardBloomlter[ 17 ],inordertokeepalowfalsepositiverate,lisontheorderofn.Forexample,inoursimulation,l=9nbits,b=6andthefalsepositiverateis1.33%.Therefore,whena=(n=logn),itisbenecialtouseBloom-lter-basedscheme.Thisisequivalenttothesituationwhenthepercentageofboundarynodesis(1=logn).Foralarge-scaleWSN,1=lognisaverysmallnumber,whichmeanstheBloom-lter-basedACkshouldbeusedinpracticewithhighprobability.Tosumup,inthissection,wedesignsomecomplementarycomponentstoBOND,tocompleteourBOND-basedCIPprotocol.Weemphasizethatthesedesignsareintuitiveandstraightforward,becausesimplicity,i.e.,minimalcommunicationandcomputationaloverhead,isourprincipaldesignobjective.However,theadvantagesofadoptingthesesimpledesignsstemfromtheBONDalgorithm.Specically,BONDneedstocheckonlyminimalnumberofconsultingneighbors,whichenablesustoutilizetwo-timerschemewithoutintroducingtoomuchcommunicationoverhead.Moreover,BONDisabletoidentifyboundarynodes,whichfacilitatesustoemployACK-basedschemetoprovidereliablesensor-to-BScommunications. 2.5ComparisonandSimulationTothebestofourknowledge,thereisnoothercoverageinferenceprotocoldevelopedforWSNswhichisascompleteasourCIP.We,however,notethatmanyinterestingideasproposedbyresearchersforWSNcoverageandself-monitoringmaybeadaptedtoinferWSNcoverage.Inthissection,weexploitthesepossiblealternativesandcomparethemwithourCIPviaboththeoreticalanalysisandsimulations.Themostimportantmetricusedinthecomparisonistheenergyconsumptionincurredbydifferentcoverageinferenceprotocols.Inaddition,weassumethatpackettransmissionsaresubjecttonoiseorcollision.Underthisassumption,wefurther 46

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denetwootherperformancemetrics.Therstoneisthefalsealarmprobability,denedastheprobabilitythatanon-boundarysensornodeisfalselydiagnosedasaboundarynode.ThismayhappenifconsecutiveEUPsfromasensornodegetlostduetocollisionand/ornoise.Theothermetricistheresponsedelay,denedasthetimefromwhenasensornodebecomesaboundarynodetowhenthiseventislocallydetected.Apparently,thelatertwometricsareconictinginessenceforatimeout-basedcoverageinferenceprotocollikeourCIP.Inparticular,toachieveasmallerfalsealarmprobability(i.e.,moreaccurateboundarydetection)desiresalargertimeoutvaluewhich,inturn,wouldresultinalongerresponsedelay,andviceversa.Therefore,wewouldcomparethefalsealarmprobabilitiesofdifferentcoverageinferenceprotocolsforagivenresponsedelay,orequivalentlycomparetheirresponsedelaysforagivenfalsealarmprobability.TheevaluationphilosophyandsimulationsettingsaregiveninAppendixA .Inordertomimicindependentandrandomnodefailures,wesimulatemultiplesensornetworksbyvaryingthenumberkofneighborsforeachnode(orequivalently,thenodedensity)throughNS2networksimulator[ 113 ].Tofacilitatethepresentation,wealsoclassifythepossiblecoverageinferenceprotocolsintotwocategories,namely,boundary-node-basedapproachesandaggregation-basedapproaches,basedonhowthecoverageinformationisgatheredandreportedtotheBS. 2.5.1Boundary-Node-BasedApproachesApproachesinthiscategoryaretondboundarynodesrstandthentransmitsuchinformationtotheBSinawaysimilartowhatwasdescribedinSection 2.4 .Inwhatfollows,wefurtherclassifytheseapproachesaccordingtotheboundarynodeidenticationmethodstheyadopted. 2.5.1.1Polygon-basedschemesIn[ 40 53 157 ],Voronoidiagram(VD)isusedforboundarynodedetection.Brieyspeaking,theVDofanodesetV,isthepartitionoftheEuclideanspaceintopolygons, 47

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Figure2-6. EnergyconsumptionfortheLVP-andVP-basedschemes. calledVoronoipolygons(VPs)anddenotedbyVor(si)forsi2VsuchthatallthepointsinVor(si)areclosertosithantoanyothernodeinV.AccordingtotheclosenesspropertyofVPs,ifsomeportionofaVPisnotcoveredbynodesinsidetheVP,itwillnotbecoveredbyanyothernode,whichimpliesacoveragehole.Therefore,itisclaimedin[ 40 53 157 ]thateachnodecanlocallycheckwhetheritisonthecoverageboundaryundertheassumptionthatVPscanbederivedlocally.However,ithasbeenshownthatingeneralVPscannotbelocallycomputed[ 169 ].Intuitively,ourLVP-basedBONDwillhavesmallercommunicationoverheadorequivalently,smallerenergyconsumptionthantheVP-basedschemessinceLVPscanbelocallycomputed.Inwhatfollows,weprovethisintuitioninaformalway. Theorem2.4. Ifthereexistboundarynodes,thecostsoftheNEP-basedandLVP-basedalgorithmsarealwayssmallerthanthecostoftheVP-basedones.Theproofofthetheoremdependsonthefollowinglemma: Lemma2. Foranysi2V,theVPVor(si)canbelocallycomputedifandonlyifClust(si)cancompletelycovertheplaneR2(orAl,whentheinformationof@Alisavailable),i.e.,Cover(si)=R2(orCover(si)\Al=Al). Proof. Letd=maxkv)]TJ /F3 11.955 Tf 11.96 0 Td[(sikforanyv2Vor(si).SinceVor(si)isaconvexset,thend=1ifVor(si)isinnite,otherwisedisthedistancefromavertexofVor(si)tosi. 48

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Vor(si)canalsobecomputedinasimilarwayasLVPwithsetVasinput.Specically,werstcomputeLVor(si)asthetentativeVPofsiandthenrenethetentativeVPiteratively.Ineachiteration,weaddone-more-hopinformationaboutnodepositions.WecandeterminethattheconstructionofVor(si)iscompletedwhenallthenodesinDisk(si,2d)havebeencounted.Therefore,Vor(si)canbelocallycomputed,whichimpliesthat2drcordrsandthusguaranteesthecompletecoverageofVor(si).Sincethisholdsforallsi2V,wecanensurethecompletecoverageoftheplane.FromTheorems 2.1 and 2.2 ,anodesetVcancompletelycoverR2ifandonlyifLVor(si)isfullycoveredbyDisk(si,rs)foranysi2V.FromLemma 1 ,thisimpliesthatVor(si)=LVor(si)foranysi2V.Therefore,Vor(si)canbelocallycomputedbysijustasLVor(si). Therefore,whenthereareboundarynodes,itisimpossibletocomputeallVor(si)'slocallybasedononlyone-hopinformation.Sincemulti-hopcommunicationsareunavoidable,theenergyconsumptionoftheVP-basedapproacheswillbehigherthanourLVP-basedBOND.OnlywhenthenodedensityissohighthattheROIiscompletelycovered(notconsideringtheROIborder),isthecostoftheVP-basedapproachesequaltothatofours.However,inthiscase,thereisnoneedforcoverageboundarydetectionatall.SoTheorem 2.4 guaranteesthatwhenboundarydetectionalgorithmsarehelpful,thecostofourBONDisdenitelysmallerthantheVP-basedones.Weanswerthequestionofhowsignicantthecostsavingsarebysimulations.Tofocusoncomparingtheenergyconsumptionoftwoschemes,wesetTEUP=1s,andC0=C1=40sand80s,andC2=0forourBOND.Forafaircomparison,theVP-basedschemeusesthesametimerC0forone-roundboundarynodedetection.Wesetaverylargetimeoutvaluetomakesurethatfalsealarmprobabilityisverysmallandthuscanbeneglectedhere.ThepacketsizeofEUPis64bytes(noNQPhere),andanycontrolpackettransmittedfortheVP-basedschemetoreconstructVPsisof64bytesaswell. 49

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Figure2-7. Perimeter-basedboundarynodedetectionapproaches.(a)Perimeter-coveragecheckingapproachproposedin[ 69 ].Thesolidcurverepresentstheportionofperimeterofsensingdiskcoveredbyneighbornodes.(b)Crossing-coveragecheckingapproachproposedin[ 62 174 ].Solidandopentrianglesrepresentcoveredanduncoveredcrossings,respectively. Figure 2-6 showstheaveragenodeenergyconsumptionfortheVP-basedandtheLVP-basedschemesasafunctionofk.WecanseethatenergyconsumptionofourLVP-basedBONDonlyslightlyincreaseaskbecomeslarger,asthereceptionenergyconsumptionbecomeslargerwiththeincreasingnumberofneighbors.Inaddition,theenergysavingsofourLVP-basedBONDarequitesignicantcomparedtotheVP-basedscheme.Wecanalsoobservethatthedifferencebetweentwoschemesbecomessmallerwiththeincreaseofk.Thereasonisthatthenumberofhopsneededtoreach2dforVPcomputingwillbecomesmallerwiththeincreaseofk.Inparticular,whenk=50wheretheROIisfullycoveredwithprobability0.9999,VPscanbelocallycalculatedasLVPs,sotherewillbenodifferencebetweenthesetwoschemes.Therefore,comparedtotheVP-basedscheme,ourBOND-basedCIPcanachieveremarkableenergysavingswhenthefunctionalnodesaresparselydeployed(4.5
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necessaryformostsituationsweareinterestedin,andLVPitselfisenoughforboundarynodedetection. 2.5.1.2Perimeter-basedschemesTherstrealisticlocalizedboundarynodedetectionalgorithmisproposedin[ 69 ],whichisbasedontheinformationaboutthecoverageoftheperimeterofeachnode'ssensingdisk.Itcanbeshownthatnodesiisaboundarynodeifandonlyifthereexistsatleastonepointv2@Disk(si,rs)whichisnotcoveredbyanysj2Neig(si)(cf.Figure 2-7 (a)).Basedonthiscriterion,analgorithmwiththecomplexityO(klogk)isdesignedin[ 69 ]tolocallycheckwhetheronenodeisaboundarynode,wherekisthenumberofneighbors.Acrossing-coveragecheckingapproachproposedin[ 62 174 ]simpliesthepreviousperimeter-coveragecheckingapproachbyjustcheckingsomespecialpointscalledcrossingsontheperimeter.Acrossingisdenedasanintersectionpointoftwoperimetersofsensingdisks.Anodesiisaboundarynodeifandonlyifthereexistsatleastonecrossingv2@Disk(si,rs)\@Disk(sj,rs)whichisnotcoveredbyanyothersk2Neig(si))-256(fsjg.Figure 2-7 (b)showsanexamplewherecisacrossingdeterminedbytwoperimeters@Disk(si,rs)and@Disk(sj,rs),whichiscoveredbythethirdsensingdiskofnodesk.Forsimplicity,belowwewilljustcompareourBOND-basedCIPwiththecrossing-coveragecheckingapproach(denotedbyCROSS).OurBONDandCROSScanbothprovidetrulylocalizedboundarynodedetectionwithoperationaldifferenceintheneighborhoodmonitoringphase.Inparticular,CROSSrequireseachnodetocheckthepositionsandstatusofallitneighbors,whichisquiteinefcientwhensensornodesaredenselydeployed.Thissituationbecomesworseeverytimewhenanodedies,asallitsneighborsneedrecheckthecoverageoftheirperimetersorcrossings.Incontrast,ourBOND-basedCIPonlyhaveconsultingneighborsperformboundarynodedetection.Whensensornodesaredenselydeployed,fromLemma 1 ,wehaveLVor(si)orTLVor(si)!Vor(si),anditiswellknownincomputationalgeometrythatunderthehomogeneousSPPP,theaveragenumberof 51

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Figure2-8. Averagenumberofneighbornodesneededforthecrossing-coveragecheckingapproachandourBOND. verticesofVor(si)is6[ 123 ].Therefore,whenthenodedensityishigh,eachnodeonaverageonlyhas4to6consultingneighbors.Figure 2-8 showstheaveragenumberofneighborsneededfortheCROSSandourBONDtodetectboundarynodesasafunctionofk.Itisofnosurprisetoseethatwhenthenodedensityincreases,thenumberofnodesneededremainsconstantforBOND-basedCIPwhileincreasingdramaticallyforCROSS.Thismeansthat,incontrasttoourBOND-basedCIP,CROSSwillincurasignicantoverheadattheinitialstageofWSNswheresensornodesarenormallydenselydeployedtoprovideadequateredundancyandfault-tolerance.NowwecompareBOND-basedCIPwithCROSSregardingtheirtradeoffsamongtheupdatingtimeintervalTEUP,responsedelayandfalsealarmprobability(denotedbyPr[FA])viasimulations.SinceCROSSneedstheinformationofallneighbors,itcanonlyusethesametimerC0forallneighbors.Therefore,theresponsedelayofBOND-basedCIPandCROSScanbemeasuredbyC1+C2andC0,respectively.ThepacketsizesofEUPandNQPareboth64bytes.OurobjectivehereistoselectoptimalsystemparameterstomeetacertainperformancerequirementoneithertheresponsedelayorPr[FA]. 52

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Figure2-9. SimulationresultswithTEUP=10s,Pr[FA]0.01. Figure2-10. Simulationresultswithresponsedelay40s,Pr[FA]0.01. WerstconsiderthesmallestresponsedelaycanbeachievedfordifferentschemeswhenTEUP=10sandPr[FA]0.01.SimulationresultsinFigure 2-9 showthat,whentheaveragenumberofneighbors(k)increases,theresponsedelayforCROSSdramaticallyincreasesduetomorepacketcollisionsinthesharedwirelesschannel,butonlyslightlyincreasesinourBOND-basedCIP.Figure 2-10 showsthepowerconsumptionfordifferentschemeswhenPr[FA]0.01andtheresponsedelay(C1+C2orC0)isboundedby40s.Itcanbeseenthat,inordertokeepPr[FA]0.01,CROSShastoadoptasmallvalueofTEUP,whichwoulddramaticallyincreasethepowerconsumption.Theenergyconsumptionofour 53

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BOND-basedCIPslightlyincreaseswithkbecausethenumberofconsultingneighborswillnotincreasewithkandtherelatedNQPswillbeusedwithverysmallprobability.FromFigure 2-9 andFigure 2-10 ,wecanalsoobservethatwhenfunctionalnodesaresparselydeployed(k15),CROSSalmosthasthesameperformanceasourBOND-basedCIP.Tosumup,theVP-basedapproachesonlyperformwellwhenfunctionalnodesaredenselydeployed;theperimeter-basedapproachesonlyworkwellwhenfunctionalnodesaresparselydeployed;andonlyourBONDworksequallywellinbothcases. 2.5.2Aggregation-BasedApproachesAggregation-basedapproachesarequitedifferentfromourBONDinthateachnodeactivelycommunicatewiththeBSnomatterwhetheritisaboundarynodeornot.ThereforeweneedtocomparethoseschemestoourBOND-basedCIPasawholeinthissubsection.Inwhatfollows,weclassifytheseapproachesintonaiveonesandspatialones. 2.5.2.1NaiveschemesThemoststraightforwardschemerequiresthateachnodeperiodicallysendEUPs(keep-alivemessages)toinformtheBSaboutitsexistencesothattheBScanalwayslearntheconnectedcoverageoftheWSN.IftheBSdoesnotreceivetheupdateinformationfromaparticularnodeinapre-determinedtimeinterval,itcaninferthatthisnodeisdead.Thisapproachobviouslysuffersfromsignicantcommunicationoverheadandthusenergyconsumption.Analternativetothenaiveschemeistoleteachnodereportthepositionsofitsdeadneighbors,inwhichnodescanlocallycooperatetoensurethateachdeadnodeisreportedonce.However,thisschemeisunlikelytobetheoptimalonebecausethedeathofsomenodesdoesnotnecessarilyimplythechangeofconnectedcoverageespeciallywhentheWSNisdenselydeployed.Inaddition,theremightberedundant 54

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Figure2-11. IllustrationofSAB.(a)TheroutingtreeforthespatialaggregationisconstructedwiththerootastheBS.(b)NodeAaggregatescoverageinformationbycombiningpolygonsfromitschildrenBandC.Notethatonlythesoliddotsarerealboundarynodes.Iftheopendotappearsintheaggregationresults,itrepresentsredundantinformationandleadstoenergywaste. informationtransmittedtotheBS,whichmeansthatthisschemeisalsonotenergyefcient. 2.5.2.2Spatialaggregation-basedschemes(SAB)Sincethecoverageinformationishighlyspatially-correlated,anaturalwaytoimprovetheenergyefciencyoftheabovenaiveschemeoritsalternativeistoperformspatialaggregationatintermediatenodestoreduceredundanttransmissions.AlmostalltechniquesforspatialaggregationrequiretheconstructionofaroutingtreeforpropagatingdatafromsourcenodestotheBS[ 91 ].Oncetheroutingtreeisestablished,eachnodeutilizestheroutingtreetondapathtotheBS.Asimplemethodforconstructingtheroutingtreeisasfollows.TheBSbroadcastsaninitializationmessageintothenetwork,whichcontainshopcountspecifyingthedistancefromtheBS.AllnodesthatheartheinitializationmessagewillselecttheBSastheirparent,increasethehopcountbyone,andthenrebroadcastthemessage.Themessage 55

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willpropagatedownthenetworkuntileverynodehasestablishedasenderasitsparentnodeleadingtowardtheBS(refertoFigure 2-11 (a)foranexample).Thecoverageinformationineachnodeisrepresentedbyapolygon.LetVchilddenoteitself,itschildren,itschildren'schildren,ThispolygoncoversallnodesinVchild,anditsverticesaretheboundarynodesofVchild.Toobtainthispolygon,eachnodeonlyneedstoreceivethepolygonsofitschildrenandaggregatesthemintoanewonewhichissentalongtheroutingtreetowardstheBS.Someboundarynodesreportedfromthechildrenmaybedeletedinthisstepbecausetheparentnodeshavemoreinformationaboutthecoverageandthusmayndthatthosenodesarenotrealboundarynodes.PleaserefertoFigure 2-11 foranillustrationofthisprocess.Thiskindofapproachisusedbye-Scan[ 179 ],whichfocusesonmonitoringtheresidualenergyofsensornodes,andisobars[ 64 ],whichfocusesonadvancedaggregationtechniquesforhighlyspatially-correlatedsensingdata.Basicallyspeaking,theenergyefciencyofSABdependsonthetimeintervalbetweentwosuccessiveactivereportsofindividualnodes.Inpractice,itisdifculttoadaptivelytunethisparametertoachievethebesttradeoffbetweeninformationfreshnessandenergyefciency.ItispossibletoadaptSABtoamorepassivescheme.Inparticular,onlywhenanodendsthatitsaggregationpolygonchanged,doesitupdatethepolygontoitsparent.However,thereisstillroomtofurtherimprovetheenergyefciency.Notethat,toreconstructtheconnectedcoverageimageattheBS,theonlyusefulinformationisthepositionsofboundarynodes.AsshowninFigure 2-11 (b),inthislayeredaggregationscheme,aggregationresultsfromthe(i+1)-hopnode(e.g.,BorC)willcontainmoreredundantinformation(non-boundarynodes)thanthoseofthei-hopnode(e.g.,A),andonlythepolygonaggregatedattheBSisredundant-free.Incontrast,ourCIPdirectlygetstherealboundarynodesfromtherststepinsteadofremovingtheredundancylayerbylayer. 56

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Figure2-12. EnergycostratioofSABandCIPtonaivescheme. Wecomparethenaiveschemeinwhichperiodickeep-alivemessagesneedbesent,SAB,andCIPunderthehomogeneousSPPPmodelthroughsimulations.WesetTEUP=1sandC0=C1=40sor80s,andC2=0forourBOND-basedCIP.SinceC1ismuchgreaterthanTEUP,thefalsealarmprobabilityisverysmallandcanbeneglectedhere.WealsoassumethatthepacketsizeofEUPis64bytes(noNQPhere),andthatitrequires32bitstorepresentaposition.Therefore,inSAB,whenachildsendsapacketaboutitspolygontohisparentnode,andthispolygonhasmvertices,thepacketsizewillbe64+4(m)]TJ /F6 11.955 Tf 12.83 0 Td[(1)bytes.Forafaircomparison,alltheschemesusethesameroutingtreeasshowninFigure 2-11 .LetRdenotetheenergycostratiowhichisdenedastheratiooftheenergyconsumptionofSABorBOND-basedCIPtothatofthenaivescheme.Figure 2-12 plotsRversusdifferentnodedensity=k=r2c,wherekistheaveragenumberofneighborspernode.Aswecansee,whenk4,almosteverynodeconnectedtotheBSisaboundarynode,andthenaiveschemeisthebest(R<1forbothBOND-basedCIPandSAB).However,thisisthesituationtheWSNtriestoavoid,underwhichtheWSNhasalreadycollapsed.Inaddition,whenk5,theRofBOND-basedCIPgrowsexponentially,whichshowsthesignicantenergysavingsofourschemeoverboththenaiveschemeandSAB.AlsonotethatwhentheROIisalmostfullycovered,thecostoflocalbroadcastsisnegligibleas 57

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comparedtoend-to-endcommunicationsfromsensornodestotheBS.Therefore,itisofnosurprisetoseethatR!1forBOND-basedCIPwhenk25.RecallthatalthoughBOND-basedCIPhasbetterperformancecomparedtoSAB,itisbasedontheassumptionthatrc2rs.Ifthisassumptiondoesnothold,boundarynodescannotbelocallyidentied(asweprovedinSection 2.3.6 ),andalladvantageswillbelost.However,forSAB,itcanbeappliedtoallvaluesofrc=rs,whichgivesanothermotivationforustopresentSABhere. 2.6ExtensionstoCIPInthissection,weextendourBOND-basedCIPbyconsideringlocationerrorsandenergydepletion,whichshowstheexibilityofBONDanditsabilitytodealwithsomepracticalconsiderations 2.6.1Location-Error-TolerantCIPSofarwehaveassumedthateachnodeknowsitsaccuratelocation.OurCIPcanalsobeextendedtotolerateboundedlocationerrors.Inthissubsection,weassumethatthelocationerror(denedasthedistancebetweentheactuallocationofanodeanditsestimatedlocation)isupper-boundedby.Wethenhavethefollowingtheorem. Theorem2.5. Ifthelocationerrorisupper-boundedby,andagivennode,e.g.,si,isaninteriornodewhenallnodesareattheirestimatedlocationsandeachnodeusesasensingrangers)]TJ /F7 11.955 Tf 12.1 0 Td[(,nodesimustbeaninteriornodewhenallnodesareattheiractuallocationsandeachnodeusestheactualsensingrangers. Proof. Weprovebycontradiction.Letaanda`representtheactualandestimatedlocationsofpointa,respectively.Supposethatnodesiisaboundarynodewiththeactualsensingrangerswhenthereisnolocationerror.TheremustexistapointpinLVor(si)thatisnotcoveredbyDisk(si,rs).Ontheotherhand,pmustbecoveredbyDisk(si`,rs)]TJ /F7 11.955 Tf 13.02 0 Td[()whenpisstillinLVor(si`)orpisinLVor(sj`)forsj`2Neig(si`).NotethathereLVor(si`)andLVor(sj`)arecalculatedwithestimatedlocationsandthesensingrangers)]TJ /F7 11.955 Tf 12.39 0 Td[(.Wethereforehaveksi`)]TJ /F3 11.955 Tf 12.39 0 Td[(pkrs)]TJ /F7 11.955 Tf 12.39 0 Td[(orksj`)]TJ /F3 11.955 Tf 12.39 0 Td[(pk
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Figure2-13. Voronoi-diagrambasedcoverageholeprediction. sj`2Neig(si`).Sinceksi)]TJ /F3 11.955 Tf 11.08 0 Td[(si`kandksj)]TJ /F3 11.955 Tf 11.08 0 Td[(sj`k,fromthetriangleinequalitywehave:ksi)]TJ /F3 11.955 Tf 11.87 0 Td[(pkksi`)]TJ /F3 11.955 Tf 11.87 0 Td[(pk+ksi)]TJ /F3 11.955 Tf 11.88 0 Td[(si`krsorksj)]TJ /F3 11.955 Tf 11.87 0 Td[(pkksj`)]TJ /F3 11.955 Tf 11.87 0 Td[(pk+ksj)]TJ /F3 11.955 Tf 11.88 0 Td[(sj`k0,theconditionthatrc=rs`2stillholds,andfromthediscussioninSection 2.3.6 ,weknowthatourBONDcancorrectlydetectboundarynodeswithrs`andestimatedlocations.BasedonTheorem 2.5 ,alltheboundarynodeswiththeactualsensingrangersandlocationswillbedetectedbyBOND.Itisworthnotingthatitispossiblethatsomeinteriornodesintermsoftheiractuallocationswillbemislabeledasboundarynodes.Therefore,ourlocation-error-tolerantCIPgivesaconservativeinferenceontheconnectedcoverage,whichisdesirableformanyWSNapplicationssuchassecurity-criticalones. 2.6.2Prediction-BasedCIPSincenodedeathcausedbyenergydepletionispredictable,itispossibletodesignprediction-basedCIPbyexploitingtheresidualenergyinformationofsensornodes.Thechallengehereisthat,thedeathofsomenodesdoesnotnecessarilyindicatethechangeofcoverageespeciallywhentheWSNisdenselydeployed.Therefore,inorder 59

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tominimizetheinformationrelayedtotheBS,weneedtodetectthenodeswhosedeathcausedbyenergydepletionwillaffectthecoverage.Werstdenehealthynodesasthosewithresidualenergymorethanathreshold.Wheneveranodendsthatitsownresidualenergyislessthanthethresholdanditisnotaboundarynode,itneedlocallybroadcastthisinformation.LetHNeig(si)bethesetofhealthyneighborsofnodesi.Whennodesindsthatitsresidualenergyislessthanthethreshold,itrstcalculatestheVPforeachofitshealthyneighbors,e.g.,sj,asfollows:Vor(sj)=\sk2HNeig(si)fsjgDom(sj,sk).NodesithencheckalltheverticesofVor(sj)(sj2HNeig(si))inDisk(si,rs).IfatleastoneofthemisonlycoveredbyDisk(si,rs),thedeathofnodesiwillcausethecoveragehole(refertoFigure 2-13 (a)foranexample),andnodesiwillreporttheeventthatitsresidualenergyissmallerthanthethresholdtotheBS.Otherwise,nodesicanconcludethatitsdeathwillnoteffectthecoverage(refertoFigure 2-13 (b)foranexample)andthusdoesnotneedtoreportitselftotheBS.Basedonthecollectedresidueenergyinformation,theBScanpredictwherethecoverageholewillemergewithhighprobability.Notethat,whenthenumberofhealthyneighborsofnodesiisk,thereareelegantalgorithmsincomputationalgeometry[ 123 ]tocalculatetheVoronoidiagramofHNeig(si)withcomplexityO(klogk),andthereareatmostO(k)verticesneedtobechecked.ThepolygonoperationsinBONDcanalsobereusedhere.Sinceonlylocalbroadcastsareinvolvedinourscheme,thecomputationalandcommunicationoverheadintroducedisrathersmall. 2.7ChapterSummaryInthischapter,weproposeacoverageinferenceprotocol(CIP)whichallowstheBStogetanaccurateandin-timemeasurementofthecurrentconnectedcoverageoftheWSN.TheCIPisbuiltuponanovellightweightdistributedboundarynodedetectionscheme(BOND).Detailedtheoreticalanalysisandsimulationstudiesshowthatboth 60

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BONDandCIParehighlyeffectiveandefcient.Asthefutureresearch,weplantoevaluatetheperformanceofBONDandCIPinrealsensorplatforms.WealsointendtofurtherinvestigateotherpotentialapplicationsofBONDinWSNssuchasloadbalancing,topologycontrol,distributedstorage,andnetworkhealthmonitoring. 61

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CHAPTER3LINKHETEROGENEITYANDDECENTRALIZEDROUTING 3.1ChapterOverviewThemodernviewoftheso-calledsmall-worldphenomenon1canbetracedbacktothefamousexperimentsbyStanleyMilgraminthe1960s[ 116 ].Hisworkshowedthatanytwopeopleintheworldcanbeconnectedbyachainof(ontheaverage)sixacquaintances,andpeoplecandelivermessagesefcientlytoanunknowntargetviatheiracquaintances.Thesmall-worldphenomenonhasalsobeenshowntobepervasiveinnetworksfromnatureandengineeringsystems,suchasthenervoussystemofthenematodewormCaenorhabditiselegans[ 1 ],foodwebs[ 161 ],theWorldWideWeb[ 6 160 ],P2Psystems[ 8 109 173 ],weboftrustforsecuritysystems[ 22 ],etc.Graphmodelstoexplainwhysocialnetworksdevelopasmalldiameter(maximumhopcountoftheshortestpaths),havebeenaroundforsometime.WhileErdos-Renyirandomgraphspossessthepropertyofhavingasmalldiameter[ 16 74 ],itiswell-knownthattheyarenotgoodmodelsforsocialnetworksbecauseoftheassumptionofindependenceoflinks[ 160 ]2.WattsandStrogatz[ 160 ]conductasetofre-wiringexperimentsongraphs,andobservethatbyre-wiringafewrandomlinksinnitelattices,theaveragepathlengthcouldbereduceddrastically,whichissmallerthanlogarithmofthenumberofvertices(nodes/points)ofthegraph.Thisleadsthemtoproposetheclassicdiscretesmall-worldmodelwhichessentiallyconsistsofalatticeaugmentedwithrandomlinksactingasshortcuts. 1Anetworkissaidtohavethesmall-worldpropertywhenthehopcountoftheshortestpathinthatnetworkisnotstronglyaffectedbyanincreaseinthenetworksize.Generally,thehopcountshouldbesmallerthanlogarithmicinthenetworksize.2Forthesamereason,Erdos-Renyirandomgraphisalsonotagoodmodelforthewirelessnetworks,see[ 63 ]foramoredetailedexplanation. 62

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ThesociologicalexperimentsofMilgramdemonstratednotonlytheexistenceofshortchainofacquaintancesbetweenstrangersbutalsotheabilityofpeopleatndingsuchchains.Whichgraphmodelshavethisproperty?Specically,whencandecentralizedroutingalgorithms(whichwewillformallydenelater)ndashortpathbetweenarbitrarysourceanddestinationnodes?ThisquestionisrstaddressedbyKleinberg[ 83 84 ]fortheclassofnitek-dimensionallatticesaugmentedwithlong-rangeconnections(shortcuts)chosenrandomlyfromthe-harmonicdistribution,thatis,along-rangelinkbetweennodesuandvexistswithprobabilityproportionaltod(u,v))]TJ /F19 7.97 Tf 6.59 0 Td[(,whered(u,v)denotestheManhattandistancebetweennodesuandv.KleinbergshowsthatthesimplegeographicgreedyroutingalgorithmbyusingonlylocalinformationcanroutemessagesbetweenanytwonodesinO(log2n)expectednumberofhopsif=kandthatthereisnoefcientdecentralizedroutingalgorithmif6=k.Notethatthereisafundamentaldifferencebetweentheexistentialdiscoveryandthealgorithmicdiscovery.Itisquitepossiblethatshortpathsexist,butthatthesecannotbefoundbyanyalgorithmusingonlylocalknowledgeofthenetwork.Forexample,Kleinberg'sresultsshowthatdecentralizedroutingalgorithmscannotndshortrouteswhen6=k,eventhoughsuchroutesindeedexistfor<2k,asdemonstratedin[ 30 ].Whileitisawellrecognizedseminalcontribution,Kleinberg'smodelisslightlyunnaturalsinceitisadiscretemodelthatassumesallnodestobelocatedonalattice,whichisoftennotthecaseintherealworld.Inthischapter,werstextendKleinberg'sresulttoamorerealisticmodelconstructedfromageometricnetwork,whereinthenodesaredistributedina2-dimensionalor3-dimensionalspaceasaspatialPoissonpointprocess(homogeneousornonhomogeneous),andtheprobabilityofanedge(link/connection)betweenapairofnodesuandvisgivenbyafunctiong()ofthedistanced(u,v),aswellasthepopulationbetweenthenodes.Suchspatialgraphmodelsandvariantsthereofarise,forinstance,inthestudyofsocialnetworksorwirelesscommunicationnetworks.Moreimportantly,weshowthatin 63

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nonhomogeneouscases,thenecessaryandsufcientconditionforgreedygeographicroutingtobeefcientisthattheprobabilityofashortcutbeingpresentfromnodeutovshouldbeinverselyproportionaltothenumberofnodeswhichareclosertouthanvis.Notethatourmodelgivesthesameshortcutprobabilitiesasmodelsinpreviousworkwhereinthenodesaredistributeduniformly.Therefore,ourworkcanalsobeappliedtohomogeneouscasesandgivesmoregeneralconditiononthenavigabilityofanygeometricnetwork.Ourresultshowsthatitisthepopulation-densitybasedshortcutdistributionwhichrelatestothenavigabilityofthegeometricnetworksratherthanthegeographic-distancebasedshortcutdistributionsuggestedinKleinberg'swork.Therestofthischapterisorganizedasfollows.InSection 3.2 ,wesurveytherelatedwork.InSection 3.3 andSection 3.4 ,weprovidethemodelofPoissonnetworksandthewaystodeterminethenetworkparameters.InSection 3.5 weprovethemainresultsinthischapter.SomeimplicationsandapplicationsofourresultstothewirelessnetworkengineeringarediscussedinSection 3.6 .Finally,weconcludethischapterinSection 3.7 3.2RelatedWork 3.2.1RelatedWorkonSocialNetworksKleinberg'sseminalworkintroducedanewthemeinthenetworkresearchliterature:navigablesmall-worldnetworks.MostimportantadvancesalongthisthemehavebeensummarizedinKleinberg'srecentaddress[ 86 ]atthe2006InternationalCongressofMathematicians.Oneinterestinglineofresearchisrelatedtothedesignofdecentralizedroutingalgorithm.RecallthatKleinbergshowsthatthesimplegreedyroutingalgorithmbyusingonlylocalinformationcanroutemessagesbetweenanytwonodesinO(log2n)expectednumberofhopsif=k.Thisboundistightenedto(log2n)laterbyBarriereetal.[ 12 ]andMarteletal.[ 110 ].Sincetheexpecteddiameterofak-dimensionalKleinbergnetworkis(logn)[ 110 ],thereisstillsomeroomforimprovingtheroutingperformance.Furtherresearch[ 44 95 109 110 ]showsthatin 64

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Figure3-1. SnapshotsondensityofwirelessusersinRomeCityon30August2006.FromRealTimeRomeProjectconductedbyMITSENSEableCityLab( http://senseable.mit.edu/realtimerome/ ). facttheO(log2n)boundoftheoriginalgreedyroutingalgorithmcanbeimprovedbyputtingsomeextrainformationineachmessageholder,whichmeanstherearesometrade-offbetweentheroutingefciencyandmemoryspacefordifferentdecentralizedroutingschemes.Inthischapter,however,wefocusonderivingtheconditionontheshortcutdistributionwhichguaranteesthenavigabilityofageometricnetworkinamoregeneralsetting.Notethattheresearchalongthislineisorthogonalandcomplementarytoourwork,inthesensethatonlywhenthegeometricgraphitselfisnavigable(whichisguaranteedbyourresults),allthoseproposedmorecomplicatedgeographicroutingalgorithmscanbeappliedinordertofurtherimprovetheroutingperformance.Kleinberg'soriginaldiscretemodelisextendedtocontinuummodelsrecently,whereinthenodesaredistributedasahomogeneousPoissonpointprocessbyFranceschettiandMeester[ 47 ],DraiefandGanesh[ 35 ].ThehomogeneousPoissonlocationofpeople(forsocialnetworks)orthenetworkdevices(forwirelessnetworks)reectsvariousirregularitiesofarealnetworkarchitecture.Thisirregularityis,however,homogeneousin[ 35 47 ],meaningthattherespectivemeandensitiesareconstantinthespace.Thisassumptionisoftennotveryrealistic.Itisenoughtotakealookata 65

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mapofthedensityofpopulationofagivenregion(refertoFigure 3-1 foranexample)torealizethatthesocialnetworkandanoptimalcommunicationnetworkthatissupposedtoreectthetrafcdemand,shouldalsobenonhomogeneous,whichindicatesthesignicanceofournewmodel(cf.Section 3.3 )whichisbasedonanonhomogeneousPoissonpointprocessofnodedistribution.3Moreover,unliketheworkof[ 35 47 ]whereintheprobabilityofshortcutsissolelydeterminedbygeographicdistance,weadoptadifferentpopulation-densitybasedshortcutformationscheme.TheideaofaddingshortcutswithprobabilityinverselyproportionaltothenumberofclosercandidatescomesfromLiben-Nowelletal.'sempiricalinvestigation[ 99 ]oftherealsocialnetworkwhichcomprisesthe1,312,454bloggersintheLiveJournalonlinecommunity(www.livejournal.com),inFebruary2004.TheyndthatKleinberg'smodelcannotbeusedtoexplainthenavigabilityoftheLiveJournalnetworkbecauseofalargevarianceinpopulationdensityacrossthespace.Theyproposeanewdensity-awaremodelofshortcutformationschemetodealwiththevarianceprobleminpopulationdensity.ThisideaisalsoimplicitinKleinberg'sworkonhisgroup-structuremodel[ 85 ],whichisbasedonpeople'smembershipingroupslikeorganizationsorneighborhoods.Thismodel,ageneralizationofhislattice-basedmodel[ 83 84 ],introducesalong-rangelinkbetweenuandvwithprobabilityinverselyproportionaltothesizeofthesmallestgroupcontainingbothuandv.Whenthegroupssatisfytwokeyproperties:amemberofagroupgmustalwaysbelongtoasubgroupofgthatisnottoomuchsmallerthang,andeverycollectionofsmallgroupswithacommonmembermusthavearelativelysmallunion,Kleinberghasproventhattheresultingnetworkisanavigablesmallworld.Themaindifferencebetweentheworkin[ 85 99 ]andthatofthischapteristhat(i)unlike 3PreviousworkshowsthatnonhomogeneousPoissonpointprocessisagoodapproximationtothedistributionofreal-lifemobilenodes,andhasbeenwidelyusedinthemodelingofthespatialdistributionofcellularphoneusers.See[ 9 ]andthereferencestherein. 66

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previousworkbasedondiscretesettings,tothebestofourknowledge,thisistherstworktoprovethenavigabilityofthegeometricnetworkswiththepopulation-densitybasedshortcutformationschemeinthegeneralornonhomogeneouscontinuumsetting;(ii)unlikepreviousworkwithfullscaleinvariance,inourmodelthescaleinvarianceiscutoffatverysmallandverylargedistances.Thereforeinourmodelthelong-rangeconnectionsforeachnodeisO(1)(cf.Section 3.4.1 )whereasinpreviousworkitisO(logn).Notethatourmodelismorerealisticinthatmaintaininglong-rangeconnectionsisobviouslymoredifcultforsocialorhybridwirelesscommunicationnetworksandshouldbeavoidedasmuchaspossible. 3.2.2RelatedWorkonWirelessNetworksSmallworldnetworkmodelisalsointroducedinthewirelessnetworkingscenarios.Forawirelessnetwork,communicationdevices(ornodes)aredistributedina2-dimensionalor3-dimensionalspace,andeachnodeisconnectedtoallitsneighborswithinsomexedradiusrc,whichiscalledthetransmissionrange.AllthoseconnectionscanbetreatedaslocalconnectionsinKleinberg'smodel[ 84 ].GuptaandKumar[ 60 ]showthatwhennnodesaredistributeduniformlyandrandomlyintheplane,theaveragenumberofhopsalongtheshortestpathbetweentworandomlychosennodes(source-destinationpair)isO(p n).Inmathematics,thiskindofnetworkistermedRandomGeometricGraph(RGG)[ 130 ],whichcanbedenotedasG(n,rc)andisobviouslynotasmallworldnetwork.Sincetheforwardingburdenisproportionaltotheaverageroutehop-length,thehopcountontheorderofnisoneofthedeterminingfactorsthatcausetheachievablethroughputforeachsource-destinationpairtoapproach0asthenetworksizengrowstoinnity[ 60 ].Inordertohandlethisvanishingthroughputeffect,manyworks[ 65 89 104 133 134 141 ]suggestaddingawiredinfrastructuretoanunstructured(ad-hoc)wirelessnetwork.Theirresultsshowthatwhenasmallnumberofwiredorwireless(e.g.,directionalantenna)long-rangeconnectionsareaddedintheadhocnetworks, 67

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theaverageroutehop-lengthcanbesignicantlyreduced,andbetterscalinglawofthroughputcanbeachieved.Hybridsensornetworkswithwiredshortcutshavealsobeenproposedintheliterature[ 29 139 141 ].Ahybridsensornetworkdiffersfromahybridadhocnetworkinthatthecommunicationinthehybridsensornetworkismany-to-one(sensor-to-sink),ratherthanmany-to-many.Basedonthespeciccommunicationtypeofsensornetworks,somewiredshortcutplacementschemeshavebeenproposed.Furthermore,theircorrespondingimprovementinenergyefciency,whichisdirectlyrelatedtothehop-length,hasbeeninvestigated.Thecommonproblemsofthoseworksinclude:theydonotaddresstheproblemofhowtondthoseshortpathsbyutilizingaugmentedshortcuts.GiventhenodesaredistributedasahomogeneousPoissonpointprocess,thegraphisconnectedonlywiththelocalconnections,andtheshortcutsaredistributedinKleinberg'sfashion,DraiefandGanesh[ 35 ]showthatnodecentralizedroutingschemecanndshortpathswithhopcountsmallerthanO(n)forany<(2)]TJ /F7 11.955 Tf 12.65 0 Td[()=6when<2,eventhoughthereexistssuchshortpathswithO(logn)hops.Therefore,itisnotnecessarythatalltheimprovementintroducedbytheinfrastructuretotheadhocnetworkscanbeachieved.Inaddition,mostoftheseworksassumethatthenodesaredistributedasahomogeneousPoissonpointprocess,whichisnotrealisticasmentionedbefore. 3.3NetworkModel 3.3.1NotationandNetworkModelWeusethefollowingnotationthroughoutthischapter: f(n)=O(g(n))meansthatthereexistsaconstantcandintegerNsuchthatf(n)cg(n)forn>N. f(n)=(g(n))meansthatf(n)=O(g(n))andg(n)=O(f(n)). f(n)=(g(n))meansthatg(n)=O(f(n)). 68

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Withhighprobability(w.h.p.)referstoaprobabilityatleast1)]TJ /F7 11.955 Tf 12.26 0 Td[((n),forafunction(n)goingto0withn!1. Prstandsforprobabilityof,andEisthecorrespondingexpectation. logdenotesthelogarithmwithbase2whilelndenotesthenaturallogarithmwithbasee. d(u,v)meanstheEuclideandistancebetweentwopointsuandv,whereu,v2R2orR3. ball(u,r)meanstheclosedballofradiusrandcenteredatpointu,i.e.ball(u,r)=fw:d(w,u)rg. Luvistheluneofuandvwhereu,v2R2orR3,i.e.,Luv=ball(u,d(u,v))\ball(v,d(u,v)). pop(A)meansthenumberofnodeslocatedintheregionAwhereAR2orR3ismeasurable. jAjistheshorthandfor2-dimensionalor3-dimensionalLebesguemeasureofameasurablesetAR2orR3.AllintegralsconsideredwillbeLebesgueintegrals. Denition3.1. [PoissonPointProcess]APoissonpointprocessPwithdensitymeasure(adiffuseRadonmeasure)isapointprocesspossessingthetwofollowingproperties: [Poissondistributionofpoint-counts]:thenumberofpointsinaboundedBorelsetBhasaPoissondistributionwithmean(B),i.e.,Pr[P(B)=m]=((B))m m!exp()]TJ /F6 11.955 Tf 9.3 0 Td[((B))form=0,1,2,... [Independentscattering]:thenumberofpointsinkdisjointBorelsetsformkindependentrandomvariables.WeassumethroughoutthischapterthattheRadonmeasurehasadensitywithrespecttoLebesguemeasure,anditcanbewrittenas[ 150 ](B)=ZB(x)dxforBorelsetsB.Thedensity(x)iscalledtheintensityfunctionofthegeneralPoissonpointprocess. 69

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Inthiswork,weconsideranetworkmodelconstructedfromanonhomogeneousPoissonpointprocessonanitesquareSn,whereineachnodeisconnectedtoallitsgeographicneighborswithinsomexedradius,aswellaspossessingrandomshortcutstomoredistantnodes.Moreprecisely: Denition3.2. [NetworkModel] [Nodedistribution]Weconsiderasequenceofgraphsindexedbyn2N.NodesfxigformanonhomogeneousPoissonprocessdenedbyDef. 3.1 ,withdensityfunctionhavingconnectedandcompactsupportwithsmoothboundary@,andmax mincwherecisaconstant. [Locallink]EachnodexiisconnectedtoallnodesxjthattheirEuclideandistanceisnotgreaterthanrn=p (cglnn)=nfor2-dimensionalcaseorrn=3p (cglnn)=nfor3-dimensionalcase.Theselinksarereferredtoaslocallinksandthecorrespondingnodesasthelocalneighborsofxi.ThegraphwithvertexsetV=fxi:1ingandedgesetconsistingsolelyoflocallinksisanonhomogeneousRGGdenotedasG(n,rn).ParametercgisaconstantcalledGEOGREEDYparameterandwillbediscussedinSection 3.4.2 [Shortcut]Fortwonodesxiandxjsuchthatd(xi,xj)>rn,thelink(xi,xj)ispresentwithprobabilityPr[(xi,xj)]=anpop)]TJ /F14 11.955 Tf 5.48 -9.68 Td[(ball(xi,d(xi,xj)))]TJ /F19 7.97 Tf 6.59 0 Td[(,wherean0and0areuniversialconstants.Theselinksarereferredtoasshortcutsandthecorrespondingnodesasthelong-rangeneighborsofxi.ThegraphwithvertexsetVandedgesetconsistingoflocallinksandshortcutsiscalledanonhomogeneousPoissonNetworkdenotedasNPN(n,rn,).Parameteranisafunctionofn,whichiscallednormalizationparameterandwillbediscussedinSection 3.4.1 .NotethathereweplacesomeconstraintsontheprobabilitydensityfunctionofthenonhomogeneousPoissonpointprocess,whichgreatlysimplifytheanalysis.Werecallthatthesupportofaprobabilitydensityfunctionisthesetofpointsinwhichithasnonzerovalue,andthattheboundary@issmoothifandonlyifitistwicedifferentiable.Wealsorequirethattheratioofthemaximumandminimumvalueofonthesupportisupper-boundedbyaconstantnumber.Thereasonofdoingsoisthatmodelinginhomogeneityisnotaneasytaskandmoreadequate,nonhomogeneousmodelsrapidlybecometoodifculttoanalyze,andwethinkallthoseconstraintsarereasonable. 70

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Here,wegivesomeintuitionbehindournetworkmodel.Thismodelhasasimplegeographicinterpretationforsocialnetworks:thenodesofasocialnetworkarethepeopleinit;twonodesareconnectedifthesetwopersonsknoweachother.Individualsknowtheirneighborswithinsomexedsmallradiusrn;theyalsohavesomenumberofacquaintancesdistributedmorebroadlyacrossthespace.Obviously,theprobabilitythattwopersonsuandvknoweachothershoulddecreaseasthegeographicaldistancebetweenthemincreases.InKleinberg'smodel,thisprobabilityissolelydeterminedbythegeographicaldistancebetweenthem(Figure 3-2 (a)),whileinourmodel,thisprobabilityisdeterminedbythepopulationinball(v,d(u,v))(Figure 3-2 (b),theshadeddisk).Intuitively,onejusticationforthiskindofshortcutdistributionisthefollowing:inordertobefriendu,vwillhavetocompetewithallofthemoreconvenientcandidatefriendsforu,i.e.,allpeoplewholiveclosertouthanvdoes.Therefore,inthischapter,whenwearemodelingthedistributionofshortcuts,weconsiderthepopulationdensitiesaswellasthegeographicaldistances.Ourconsiderationismorereasonablesincetheprobabilityoftwopersonsknoweachothershoulddecreasemorequicklywiththegeographicaldistancewhenthepopulationdensityishigh.Itisawell-knownobservationthatinthecitywithhighpopulationdensity,peopleknowtheirgeographicalneighborswithasmallprobabilitywhileinthecountry,thisprobabilitywillbemuchhigher.Forwirelessadhocnetworks,locallinksmodelthecommunicationbetweennearbynodesthroughwirelesslinks.Thiskindofdiskcommunicationmodel(withcommunicationrangern)iswidelyusedinthetheoreticalstudyofwirelessnetworks(e.g.,[ 60 ]).Shortcutscanmodelthewiredorwirelessinfrastructureaddedinthispurelyadhocnetwork.Obviously,theprobabilityofexistenceoftheshortcutshoulddecreaseasthegeographicaldistancebetweenthemincreasessincethecostoftheshortcuts(wiredinfrastructure)isproportionaltothetotallengthofthewiresdeployed[ 141 ]. 71

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Figure3-2. Navigablesmall-worldnetworkmodels.(a)Kleinberg'saugmentedlatticemodelin[ 84 ];(b)NonhomogeneousPoissonnetworkmodelusedinthischapter.Herestuffednodes,boldsolidlinesandbolddashedcurvesrepresentthelocalneighborsofu,locallinksandshortcuts,respectively.Notethatin(b),theprobabilityofobtainingashortcutfromutovisinverselyproportionaltothenumberofnodeswithintheshadeddiskball(v,d(u,v)). Thenodedensityalsoneedbeincludedintheconsiderationinordertooptimizetheplacementofthewiredinfrastructure. 3.3.2BackgroundHerewepresenttheformaldenitionsoftheconceptsusedinthischapter. Denition3.3. [Small-WorldNetwork]Forageometricnetworktobetermedasmall-worldnetwork(SWN),itsdiametershouldbeontheorderoflognoratmostpolylogarithmicinn,wherenisthenetworksize.Theobjectiveistorouteapacketfromanarbitrarysourcenodestoanarbitrarydestinationtusingasmallnumberofhops.Weareinterestedindecentralizedroutingalgorithms,whichdonotrequireglobalknowledgeofthegraphtopology. Denition3.4. [DecentralizedRouting]Adecentralizedroutingalgorithmspeciesasequenceofnodess=x1,x2,...,xk=twheretheonlyrequirementisthateachnodexi(2ik)]TJ /F6 11.955 Tf 12.35 0 Td[(1)shouldbechosenfromthelocalorlong-rangeneighborsofnodexi)]TJ /F9 7.97 Tf 6.59 0 Td[(1,andxionlyknowsthetopologyofitsneighbors(localinformation). 72

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OnespecialkindofdecentralizedroutingiscalledgeographicallygreedyroutingorGEOGREEDYinshorthand. Denition3.5. [GeographicallyGreedyRouting]Itisassumedthateachnodeknowsitslocation(coordinates)inthespace,aswellasthelocationofallitsneighbors(bothlocalandlong-range),andofthedestinationt(theheaderofthepacketcarriesthelocationoft).Ifthereisnodirectlinkfromthesourcestothedestinationt,thecurrentpacket-holderuwillforwardthepackettooneofitslocalorlong-rangeneighborsclosesttothedestinationt.Ifnoneoftheseneighborsareclosertothedestinationofthepacketthanthepacket-holderitself,thepacketwillbediscarded. Denition3.6. [NavigableSmall-WorldNetwork]ASWNiscallednavigableSWNifandonlyifthereexistsdecentralizedroutingalgorithmsuchthatthenumberofhopsformessagedeliverybetweenanypairofnodesisoforderatmostpolylogarithmicinn,wherenisthenetworksize. 3.4CharacterizationoftheParametersintheNetworkModelTherearetwoparameterscgandaninournetworkmodellefttobedeterminedinSection 3.3 .Inthissection,wewillshowhowtotunethesetwoparametersinordertoobtainthenetworkmodelwiththedesiredproperties. 3.4.1NormalizationParameteranandtheExpectedNumberofShortcutsforEachNodeSincethemaintenanceofshortcutsarecostlybothinsocialandwirelessnetworks,inourmodelweupper-boundtheexpectednumberofshortcutsforeachnodeasaconstantnumber(1). 73

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Foranynodev2V,wehave:Xxi2V,xi6=vPr[(v,xi)]=an n)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xi=11 i!=8>>>><>>>>:an(1 1)]TJ /F19 7.97 Tf 6.58 0 Td[(n1)]TJ /F19 7.97 Tf 6.59 0 Td[()if<1an(lnn)if=1an)]TJ /F9 7.97 Tf 12.69 -4.98 Td[(1 )]TJ /F9 7.97 Tf 6.59 0 Td[(1if>1Therefore,ifwerequiretheexpectednumberofshortcutsforeachnode,i.e.,PxiPr[(v,xi)],tobeaconstantnumber(1),weneedtohave: an=8>>>><>>>>:((1)]TJ /F7 11.955 Tf 11.96 0 Td[()n)]TJ /F9 7.97 Tf 6.59 0 Td[(1)if<1(1=lnn)if=1()]TJ /F6 11.955 Tf 11.95 0 Td[(1)if>1 (3) 3.4.2GEOGREEDYParametercgandtheExpectedNumberofLocalNeighborsforEachNodeWewillimplicitlyassumethatanynodecanndalocalneighborclosertothedestinationtthanitselfinthefollowingdiscussion(cf.Section 3.5 ),andcallitthesuccessfulGEOGREEDYassumption.Thisassumptionimpliesthat,GEOGREEDYalgorithmcansuccessfullyroutepacketsbetweenanysource-destinationpairsfromVVonthenetworkG(n,rn).Werstshowthatthisassumptionholdsw.h.p.ifcgischosentobesufcientlylarge. Theorem3.1. GiventhenetworkmodelNPN(n,rn,)(cf.Def. 3.2 )andtherandomlychosendestinationt,thesufcientconditionforanynodetondalocalneighborclosertotthanitselfw.h.p.isthatcg>6 (4)]TJ /F6 11.955 Tf 11.95 0 Td[(3p 3)min0.82 minfor2-Dcase;orcg>12 5min0.77 minfor3-Dcase. 74

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Figure3-3. Sufcientconditionforualwayshavingalocalneighborwclosertothedestinationvwithd(u,v)>rn.Notethatv`istheintersectionpointofthesegmentuvandthecirculusofball(u,rn).TheshadedareaisLuv`whichiscontainedinALuv(theareawhichiscirculatedbytheboldcurves). Proof:Forthe2-dimensionalcase,astraightforwardcalculationyieldsthat jLuvj=2 3)]TJ 13.15 18.04 Td[(p 3 2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(d(u,v)2,(3)whileforthe3-dimensionalcase,weobtain jLuvj=5 12)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(d(u,v)3.(3)Weset=2 3)]TJ /F16 7.97 Tf 13.25 11.35 Td[(p 3 2andfocusonthe2-dimensionalcaseinthefollowing.Theproofforthe3-dimensionalcaseisalmostthesameexceptforsetting=5 12.TwokeyobservationsfromFigure 3-3 are:(i)Thesufcientandnecessaryconditionthatnodeucanndalocalneighborclosertothedestinationvwithd(u,v)>rnisthatthereexistsatleastonenodew2VwithinALuv,whereALuvisdenedasALuv=ball(u,rn)\ball(v,d(u,v)).(ii)Luv`ALuvwherev`istheintersectionpointofthesegmentuvandthecirculusofball(u,rn).Therefore,thesufcientconditionthatnodeucanndalocalneighborclosertothedestinationvisthatthereexistsatleastonenodew2VwithinLuv`.FromEq.( 3 ),we 75

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obtain jLuv`j=r2n.(3)LetXibetheeventthatnodexidoesnothaveanylocalneighborsclosertothedestination,andletX=Sni=1XibetheeventthatthereisatleastonenodeinVwhichdoesnothaveanylocalneighborsclosertothedestination.FromEq.( 3 ),weobtainPr[Xi]= 1)]TJ /F15 11.955 Tf 11.96 16.27 Td[(ZLxixi`(x)dx!n)]TJ /F9 7.97 Tf 6.58 0 Td[(11)]TJ /F6 11.955 Tf 11.95 0 Td[(minr2nn)]TJ /F9 7.97 Tf 6.59 0 Td[(1,wherexi`istheintersectionpointofthesegmentxi`tandthecirculusofball(xi,rn).Fromtheunionbound,wecanwrite4Pr[X]n1)]TJ /F6 11.955 Tf 11.96 0 Td[(minr2nn)]TJ /F9 7.97 Tf 6.59 0 Td[(1=elnn+(n)]TJ /F9 7.97 Tf 6.59 0 Td[(1)ln(1)]TJ /F9 7.97 Tf 6.58 0 Td[(minr2n)elnn)]TJ /F9 7.97 Tf 6.59 0 Td[((n)]TJ /F9 7.97 Tf 6.59 0 Td[(1)minr2n=e(lnn))]TJ /F9 7.97 Tf 5.48 -9.68 Td[(1)]TJ /F23 5.978 Tf 7.79 3.86 Td[((n)]TJ /F23 5.978 Tf 5.75 0 Td[(1) lnnminr2n=e(lnn) 1)]TJ /F9 7.97 Tf 6.59 0 Td[(minrn p lnn n)]TJ /F23 5.978 Tf 5.76 0 Td[(12!where,towritethesecondinequality,wehaveusedthefactthatln(1+x)x.ForsuccessfulGEOGREEDYassumptiontobeholdw.h.p.,wewantPr[X]!0asn!0.Fromthenalequation,thiscanbeseentohappenifp minrn=p (lnn)=n!1,thatisrnismadetodecreasestrictlyslowerthanp (lnn)=(minn),withtheratiogoingto1asn!1.FromournetworkmodelNPN(n,rn,),wehavern=p (cg.lnn)=n.Therefore,thesufcientconditionforsuccessfulGEOGREEDYassumptioniscg>1 min. 4Foreaseofpresentation,weneglectedgeeffectsinthefollowing. 76

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FromTheorem 3.1 ,inordertosatisfythesuccessfulGEOGREEDYassumption,weneedtosetcg0.82 minforthe2-dimensionalcaseandcg0.77 minforthe3-dimensionalcase.Obviously,thesuccessfulGEOGREEDYassumptionisastrongerrequirementthanconnectivityrequirementonG(n,rn).ThisstatementisalsosupportedbyPenrose'sinvestigation[ 129 ]ontheconnectivitypropertiesofRGGinthecaseofarbitrarynodedistribution(providedthatcertaintechnicalconditionsaresatised),inwhichitisshownthatG(n,rn)isconnectedw.h.p.ifandonlyifcg>1 min0.32 minforthe2-dimensionalcase,whichissmallerthanourcgforsuccessfulGEOGREEDY.GuptaandKumar'sclassicalwork[ 60 ]showsthatinordertomaximizethethroughput,rnshouldbeassmallaspossiblewhentheconnectivityconditionissatised.Theirconclusionisforthearbitraryroutingalgorithms.Inourcase,whenGEOGREEDYisused,rnshouldbeassmallaspossiblewhenthesuccessfulGEOGREEDYassumptionissatised.Thereforewesetcg=0.82 mininthefollowingdiscussion.NotethatthecorrespondingexpectednumberoflocalneighborsNeigLoc(u)isontheorderoflnn,whichisthesameasfortheconnectivity[ 129 ].ArecentresultduetoWanandYietal.[ 156 ]showsthatforthe2-dimensionalhomogeneousG(n,rn),wherennodesaredistributeduniformlyandrandomlyoveraunitsquare,inordertosatisfythesuccessfulGEOGREEDYassumption,cgshouldbegreaterthan1=0.82.Obviously,thisisaspecialcaseofTheorem 3.1 ,whenthenormalizednodedensityisaconstant(x)=n,whichimpliesthatmin=1. 3.5NavigabilityofNonhomogeneousPoissonNetworksInthissectionwewilldemonstratethemainresultsofthischapterandthecorrespondingproofsonthenavigabilityofnonhomogeneousPoissonnetworks.Werstpresentaspecialroutingalgorithmusedinthissection.AnillustrativeexampleisgiveninFigure 3-4 .Ifthereisnodirectlinkfromthesourcestothedestinationt,themessage(packet)ispassedviaintermediatenodesasfollows.Ateachstage,thepacketcarriestheaddress(co-ordinates)of 77

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Figure3-4. ApproximateGEOGREEDYroutingalgorithm.Solidcurvesanddashedcurvesrepresentlocallinksandshortcuts,respectively.Notationx4,d=2meansitisthe4-threlaynode,withthevalueofcurrentindicatord=2. thedestinationt,aswellasanindicatorinthepacketheader.Thevalueoftheindicator,i.e.,d,isinitializedtod(s,t),thedistancebetweensandt.Supposethatthepacketiscurrentlyatnodeuandhastheindicatorvalued>rn.5Ifnodeuhasashortcuttosomenodexi2A(t,d=2),wheretheannulusA(t,d=2)isdenedasA(t,d=2)=ball(t,d=2)nball(t,d=4),thenuforwardsthemessagetoxi.Ifthereismorethanonesuchnode,thechoicecanbearbitrary.Otherwise,itforwardsthemessagetooneofitslocalneighborswhichisclosertotthanitself.Whenanodexireceivesapacket,itupdatesdwithd=2ifd(xi,t)d=2,andleavesdunchangedotherwise.Inotherwords,ifucanndashortcutwhichreducesthedistancetothedestinationbyatleastahalfbutbynomorethanthree-quarters,itadoptssuchashortcut.Otherwise,itusesalocalneighbortoreducethedistancetothedestination.Inthatsense,wecalltheroutingalgorithmdescribedaboveasapproximateGEOGREEDY 5Ifdrn,thenthenodeuwillhavecontainedtinitslocalneighborlistandwilldeliverthepacketimmediately. 78

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Figure3-5. CalculatingtheprobabilityofnodeuhavingashortcuttooneofthenodesinA(t,d=2). routingalgorithm.ThereasonforconsideringsuchanalgorithmratherthanaGE-OGREEDYdenedinDef. 3.5 thatminimizesthedistancetothedestinationateachstepistopreservetheindependence,whichgreatlysimpliestheanalysis.Notethatifagreedystepfromutakesustov(i.e.,ofallnodestowhichupossessesashortcut,visclosesttot),thentheconditionallawofthepointprocessintheball(t,d(t,v))willbeviolated.Thefactthattherearenoshortcutsfromutonodeswithinthisballbiasestheprobabilitylawandgreatlycomplicatestheanalysis.HereapproximateGEOGREEDYalgorithmgetsaroundthisproblemandhasalreadybeenprovedusefulin[ 35 ]. 3.5.1NavigabilityofNPN(n,rn,1) Theorem3.2. When=1,NPN(n,rn,1)isanavigablesmall-worldnetwork,i.e.,thereexistsadecentralizedroutingalgorithm,e.g.,approximateGEOGREEDY,toroutepacketsbetweenanysource-destinationpairschosenfromVVinO(log2n)hops.Proof:Weconcentrateonthe2-dcaseinthefollowing.The3-dcasecanbeproveninasimilarfashion.Beforeproceedingtothetechnicaldetailsoftheproof,webeginwithabriefhigh-leveloutlineoftheproofandsomeintuitionoftheanalysis.First,weclaimthattheexpectednumberofhopstakenbyapproximateGEOGREEDYbeforewereacha 79

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nodehalfwayfromthesourcetothedestinationbyutilizingtheshortcutisO(lnn);thenweshowthatafteratmostO(logn)suchhalvings,wewillreachthedestinationw.h.p.Toestablishtheclaim,werstneedtocalculatetheprobabilityofndingasuitableshortcutateachstepoftheapproximateGEOGREEDY.Wethinkoftheroutingalgorithmasproceedinginphases.Thevalueofdishalvedattheendofeachphasewhenthepacketreachesanodeusatisfyingtherelationd(u,t)2(d=4,d=2]atthersttime.Figure 3-5 illustratesthefollowing.Nodeuisthecurrentpacket-holderwiththedestinationt.DenotebyNAthenumberofnodesintheannulusA(t,d=2).Obviously NA=popAt,d 2=popballt,d 2nballt,d 4.(3)Foranyv2A(t,d=2)andv2V,thedistanced(u,v)isboundedaboveby3d=2,andthustheprobabilitythatashortcutfromuisincidentonaparticularoneofthesenodesisboundedbelowby Pr[(v,u)]anpop)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(ball(v,3d=2))]TJ /F9 7.97 Tf 6.59 0 Td[(1an pop)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(ball(t,2d). (3) Thus,conditionalonNA,theprobabilitythatuhasashortcuttooneoftheNAnodesinA(t,d=2)isboundedbelowby p(d,NA)1)]TJ /F15 11.955 Tf 11.95 20.44 Td[( 1)]TJ /F3 11.955 Tf 49.58 8.08 Td[(an pop)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(ball(t,2d)!NA.(3)FromEq.( 3 )andInequality( 3 ),weobtain p(d,NA)1)]TJ /F6 11.955 Tf 11.96 0 Td[(exp )]TJ /F3 11.955 Tf 37.91 8.09 Td[(NAan pop)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(ball(t,2d)!=1)]TJ /F6 11.955 Tf 11.96 0 Td[(exp )]TJ /F3 11.955 Tf 11.95 0 Td[(anpop)]TJ /F27 9.963 Tf 5.48 -9.68 Td[(ball(t,d=2)nball(t,d=4) pop)]TJ /F27 9.963 Tf 5.48 -9.69 Td[(ball(t,2d)!anpop)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(ball(t,d=2)nball(t,d=4) pop)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(ball(t,2d)an. (3) 80

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Ifudoesnothavesuchashortcut,thepacketispassedvialocalneighborswhicharesuccessivelyclosertot,andhencethesamelowerboundontheprobabilityofashortcuttoA(t,d=2)issatised.Consequently,thenumberoflocalstepsLuuntilashortcutisfoundisboundedabovebyageometricrandomvariablewithconditionalmean1=p(d,NA),i.e.,Pr[Lu=k]=(1)]TJ /F3 11.955 Tf 11.96 0 Td[(p(d,NA))k)]TJ /F9 7.97 Tf 6.59 0 Td[(1p(d,NA).Therefore,weobtainE[Lu]1 p(d,NA)=1 an.Sincean=(1=lnn)(cf.Eq.( 3 )inSection 3.4.1 ),weobtainE[Lu]=O(lnn).ByusingtheChernoffboundforaPoissonrandomvariable,itcanbeshownthatLu=O(lnn)w.h.p.Notethatwhen(x)=min,thesidelengthlofthesmallestsquareSnwhichcontainsnnodesisatmostO)]TJ /F16 7.97 Tf 11.21 1.67 Td[(p 2n min,whichimpliesthatthenumberofphasesisO(logn)sincetheinitialvalueofdisatmostO(p n),anddishalvedattheendofeachphase.Therefore,thetotalnumberofhopsisO(log2n),whichcompletestheproofofthetheorem.Recalltheresultsonhomogeneouscase,i.e.,nodesareuniformlydistributedinthespace,fromworksofKleinberg[ 84 ],FranceschettiandMeester[ 47 ],DraiefandGanesh[ 35 ].Theyshowthatwhentheshortcutbetweennodesuandvexistswithprobabilityproportionaltod(u,v))]TJ /F9 7.97 Tf 6.58 0 Td[(2(forthe2-dimensionalcase),decentralizedroutingalgorithmisefcient.NotethatthisisaspecialcaseofourTheorem 3.2 .Inournetworkmodel,whennodesareuniformlydistributed,i.e.,(x)=for8x2S,wehavePr[(u,v)]=anpop)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(ball(u,d(u,v)))]TJ /F9 7.97 Tf 6.59 0 Td[(1=an d(u,v)2/d(u,v))]TJ /F9 7.97 Tf 6.59 0 Td[(2.Therefore,ourworkcanalsobeappliedtohomogeneouscasesandgivesmoregeneralconditiononthenavigabilityofanygeometricnetwork.Ourresultsshowthatitisthe 81

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population-densitybasedshortcutdistributionwhichdeterminesthenavigabilityofthegeometricnetworksratherthanthegeographic-distancebasedshortcutdistributionsuggestedinKleinberg'sworkinamoregeneralsetting. 3.5.2InnavigabilityofNPN(n,rn,)When6=1Inthissubsection,weshowthat=1isalsothenecessaryconditionforgreedygeographicroutingtobeefcient. Theorem3.3. SupposethesourcesanddestinationtarechosenuniformlyatrandomfromVV.(a)When>1,theexpectednumberofhopsforroutingpacketsbetweensandtis(n()]TJ /F9 7.97 Tf 6.58 0 Td[(1)=(2));(b)When0<1,theexpectednumberofhopsforroutingpacketsbetweensandtis(n(1)]TJ /F19 7.97 Tf 6.59 0 Td[()=6).Proof:(a)Foranynodeu2V,wesortalltheothernodesinVusingthedistancetouintheincreasingorder.SincenodesaredistributedasaPoissonpointprocess,notwonodesareofthesamedistancetouw.h.p.Therefore,wecanobtainasequencexu1,xu2,,xui,,xun)]TJ /F9 7.97 Tf 6.59 0 Td[(1,wherexui2Vandxui6=ufor1in)]TJ /F6 11.955 Tf 12.46 0 Td[(1withthepropertythatd(u,xui)d(u,xui)]=O(i1)]TJ /F19 7.97 Tf 6.58 0 Td[().(3)Forrandomlychosensandt,d(s,t)>d(t,xtp n)w.h.p.Denel`=n(1=2))]TJ /F19 7.97 Tf 6.59 0 Td[((logn))]TJ /F9 7.97 Tf 6.59 0 Td[(1.Therefore,thenecessaryconditionforexistingapathoflengthnhopsbetweensandtisthatatleastoneofthehopsisashortcutoflengthl`(inhops)ormore.Let"betheeventthatsuchashortcutexists.ByEq.( 3 )andtheunionbound,weobtain: Pr["]=O n n(1=2))]TJ /F19 7.97 Tf 6.58 0 Td[( logn!1)]TJ /F19 7.97 Tf 6.58 0 Td[(!.(3) 82

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Itisobviousthatthisprobabilitytendsto0asn!1if<()]TJ /F6 11.955 Tf 12.14 0 Td[(1)=(2),whichimpliesthatif<()]TJ /F6 11.955 Tf 12.45 0 Td[(1)=(2),thentheprobabilityofndingapathwithfewerthannhopsbetweensandttendstozeroasn!1.(b)Dene=(1)]TJ /F7 11.955 Tf 12.45 0 Td[()=6.LetUdenotethesetofnodeswithindistanced(t,xtn)oft.Therefore,jUj=n+1.Forrandomlychosensandt,itisclearthatsisnotinUw.h.p.,andforanyv2U,d(s,v)>d(v,xvp n)w.h.p.Supposenowthatthereisadistributedroutingalgorithmwhichcanndapathfromstotinfewerthannhops.Denotebys=x0,x1,,xm=t,thesequenceofnodesvisitedbytheroutingalgorithm,withm
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3.6ApplicationstoWirelessAdHocNetworks 3.6.1DoestheDistributionofShortcutsCount?Theorem 3.2 producesapositiveresult,whichshowsthatGEOGREEDYroutingalgorithmisefcientforourNPN(n,rn)inthatityieldstheshortpathwithO(polylogn)hopslocally.FromtheproofofTheorem 3.2 ,wecanseethatthenavigabilityofNPN(n,rn)comesfromthespecialdistributionpatternofshortcuts.Inthisandthenextsubsection,wewillfurtherinvestigateonthisissuebyprovidingsomecounterexampleswithslightlydifferentshortcutpattern,whichyieldsshortpathswithO(polylogn)hopswhilethereexistsnodecentralizedroutingalgorithmsforndingthem.Example1:Inthisexample,thenodedistributionandtheformulationoflocallinksarethesameasinthenetworkmodeldenedinDef. 3.2 .TheonlydifferenceisthattheshortcutsareformedasanErdos-Renyirandomgraph.Moreprecisely,shortcutsarepresentbetweeneachpairofnodeswithprobabilitypn,independentofallothershortcuts.ThegraphwithvertexsetVandedgesetconsistingsolelyofshortcutsisanErdos-Renyirandomgraph(orBernoullirandomgraph)denotedasB(n,pn)[ 16 74 ].Wecallthisnewnetworkmodelthecombinedrandomnetwork,denotedasCG(n,rn,pn),whichisthecombinationofnonhomogeneousRGGG(n,rn)andB(n,pn).InordertomakethismodelcomparabletoourNPN(n,rn),wesetpn=O(1 n),therefore,thetotalnumberofshortcutsisonthesameorderofourNPN(n,rn).Inthefollowingtwotheorems,wewillshowthat,forCG(n,rn,pn),w.h.p.thereexistsshortpathsbetweeneverypairofnodeswhoselengthsareboundedbyapolynomialinlogn.However,thereisnowayforadecentralizedroutingalgorithmtondtheseshortpaths. Theorem3.4. ThediameterofCG(n,rn,pn)isw.h.p.(logn).Proof:ForErdos-RenyirandomgraphB(n,pn),wehavethefollowingwellestablishedresults[ 16 74 ]:(i)ThediameterofaconnectedB(n,pn)isoforderlogn lognpn. 84

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(ii)Ifpn=c n,c>1,theuniquegiantcomponentofsizeO(n)emerges.(iii)LetI(n)bethesetofnodesnotinthegiantcomponent,thentheexpectedsizeofI(n)isE[jI(n)j]=x() nwherex() isthefractionofnodesnotinthegiantcomponent,andx()=P1k=1kk)]TJ /F23 5.978 Tf 5.76 0 Td[(1 k!(2e)]TJ /F9 7.97 Tf 6.58 0 Td[(2)k,where=c=2>1=2.ForCG(n,rn,pn),theexistenceofgiantcomponentinB(n,rn)isensuredbysettingpn=c n,c>1.From(i),inordertoprovethetheorem,weonlyneedtoshowthatthereisnoshortestpathlongerthanlogn lognpninCG(n,rn,pn)fromnodesnotinthegiantcomponentofB(n,pn).Anecessaryconditiontohaveapathlongerthanlogn lognpnistohavelogn lognpnnodesfromI(n)inanareaofsizelessthanorequalto(logn lognpn)r2n=cglog2n nlognpn.Forxi2I(n),letNibethenumberofnodesfromI(n)thatareinanareaofcglog2n nlognpnwhichxibelongsto.LetN=maxfNi:xi2I(n)g.SinceE[Ni]=E[jI(n)j]cglog2n nlognpn=Olog2n nn!1)166(!0,itfollowsthatPr[N>logn lognpn]n!1)166(!0andsothediameterofCG(n,rn,pn)isofthesameorderasthediameterofthegiantcomponentofB(n,pn=c=n). Theorem3.5. ForCG(n,rn,pn),supposethatthesourcesanddestinationtarechosenuniformlyatrandomfromthenodesetV.Then,thenumberofhopsformessagedeliveryinanydecentralizedalgorithmexceedsnw.h.p.,forany<1=3.Proof:Weprovebycontradiction.Supposethereisadecentralizedroutingalgorithmwhichcanndapathfromstotinfewerthannhops.Denotethesequenceofnodesonthispathbys=x0,x1,...,xm=t,withmn.Fix2(,1=2)anddeneC=C(t,n)tobethecircleofradiusncenteredatt.Itiseasytoshowthat,forany>0,thedistanced(s,C)fromstothecircleCislargerthann(1=2))]TJ /F19 7.97 Tf 6.59 0 Td[(w.h.p.(notethatthesidelengthlofsquareSscalesas(p n)).Thentheremustbeashortcutfromatleastoneofthenodesx0,x1,...,xm)]TJ /F9 7.97 Tf 6.59 0 Td[(1tothesetC.Indeed,ifthereisnosuchshortcut,thentmustbereachedstartingfromsomenodeoutsideCbyusingonlylocallinks.Sincethelengthofeachlocallinkisatmostrn=p (cglogn)=nandthenumberofhops 85

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isatmostn,thetotaldistancetraversedbylocalhopsisstrictlysmallerthann(forlargeenoughn,bytheassumptionthat>),whichresultsinacontradiction.Wenowestimatetheprobabilitythatthereisashortcutfromoneofthenodesx0,x1,...,xm)]TJ /F9 7.97 Tf 6.59 0 Td[(1tothesetC.ThenumberofnodesinthecircleC,denotedbyNC,isupper-boundedbyaPoissonrandomvariablewithmeanmaxn2,soNC1andtheunionbound,weobtainPrh9shortcutbetweenuandCNCcanbechosenarbitrarily.Inparticular,wecanchoosesothat+2)]TJ /F6 11.955 Tf 12.62 0 Td[(1isstrictlynegative,inwhichcasetheconditionalprobabilityofashortcuttoCgoestozeroasn!1.SincePr[NCmax4n2]alsogoestozero,itisclearthattheprobabilityofndingans)]TJ /F3 11.955 Tf 12.53 0 Td[(troutewithfewerthannhopsalsogoestozero,whichresultsinacontradiction.Theorem 3.5 showsthatthedistributionofshortcutsdoesaffectthenavigabilityofgeometricnetworks.TheonlydifferencebetweenCG(n,rn,pn)andNPN(n,rn)isthat,inCG(n,rn,pn),shortcutsareuniformlydistributedoverthenodepairsfromVV.ThismakesCG(n,rn,pn)completelyinnavigable,eventhoughCG(n,rn,pn)hasthesamenumberofshortcutsandtheorderofdiametersasNPN(n,rn).Theauthorsof[ 133 134 ]suggesttouniformlyandrandomlyaddsomewiredshortcutsinwirelessadhocnetworksinordertoincreasetransportcapacity.Eventhoughtheirresultsshowthesignicantimprovementboththeoreticallyandexperimentally, 86

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theyfailtoconsiderthealgorithmicaspectoftheirscheme.Ourresultsinthischaptershowthatthereisnodecentralizedalgorithmthatcanachievethosebenetsfromtheshortpathsexistinginthenetwork. 3.6.2DoesAddingMoreShortcutsHelp?Intuitively,whenweaddmoreshortcutsinthenetwork,asmallerdiameterofthenetworkcanbeobtained,whichimpliesthattheshortestpathswithsmallerlengthwillemerge.However,whenweconsiderthenavigabilityofnetworks,doesaddingmoreshortcutsreallyhelp?Example2:GivenanNPN(n,rn),recallthattheexpectednumberofshortcutsforeachnodeisO(1).NowweaddmoreshortcutsonthisNPN(n,rn).TheprincipleofthisprocedureisthateachnodewillhavetheexpectednumberofshortcutsoforderO(logn),renderingthedistributionofshortcutsbetweentwonodesuniform.Therefore,originalNPN(n,rn)willbetransformedintoCG(n,rn,pn)withpn=O(logn n).FollowingtheproofprocedureofTheorem 3.5 ,itcanbeshownthatthenewnetworkbecomesinnavigable6,eventhoughcomparedtooriginalNPN(n,rn),nolinkshavebeendeletedandonlynewshortcutsareadded.Weaddasignicantamountofnewresource:infact,thenumberofnewshortcutsislogntimesthatofoldones,however,theperformanceofthenetworkdegradesgreatly:theshortestpathdiscoveredbydecentralizedalgorithmisontheorderofnwhileitislog2nfortheoriginalnetwork.Thisexampleshowsthatitisthedistributionpatterninsteadoftotalnumberofshortcutsthataffectsthenavigabilityofgeometricnetworks.Inmostcases,thenumberofshortcutsisproportionaltothecostofthesystem.Therefore,weneedtobevery 6Theorem 3.5 showstheinnavigabilityofCG(n,rn,pn)withpn=O(1 n).Herewehavepn=O(logn n),whichismuchlargerbutwillnotchangethetechnicalessenceoftheproofofTheorem 3.5 ,andhenceCG(n,rn,pn)isstillinnavigable. 87

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carefulwhenplanningthenetwork,duetothepossibilitythatmoreshortcutsmayleadtoworsenetworkperformance. 3.7ChapterSummaryInthischapter,weextendKleinberg'smodel[ 84 ]toamorerealisticmodelconstructedfromanonhomogeneousPoissonpointprocess,whereineachnodeisconnectedtoallitsneighborswithinsomexedradius,aswellaspossessingrandomshortcutstomoredistantnodes.Moreimportantly,weshowthatinnonhomogeneouscases,thenecessaryandefcientconditionforgreedygeographicroutingtobeefcientisthattheprobabilityofashortcutbeingpresentfromnodeutovshouldbeinverselyproportionaltothenumberofnodeswhichareclosertouthanvis.Notethatourmodelgivesthesameshortcutprobabilitiesasmodelsinpreviousworkwhereinthenodesaredistributeduniformly,therefore,ourworkcanalsobeappliedtohomogeneouscasesandgivesmoregeneralconditiononthenavigabilityofanygeometricnetwork.Tothebestofourknowledge,thisistherstworktoprovethisresultforthegeneralornonhomogeneouscontinuumsetting. 88

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CHAPTER4SCALINGLAWSFORLARGE-SCALEMANETSWITHNETWORKCODING 4.1ChapterOverviewOnedistinctcharacteristicofwirelessmobileadhocnetworks(MANETs)isthat,besidestransportingdatathroughmulti-hopconnectedpathsbetweenthesourceanddestination,packetscanalsobedeliveredthroughthephysicalmobilityofrelaynodeswhichiscalledstore-carry-and-forwardparadigmintheliterature[ 41 ].GrossglauserandTse[ 58 ]haveshownthatsignicantgainsinper-nodethroughputcanbeobtainedbyexploitingthisparadigm.Inparticular,theyproposeda2-hoprelayingscheme,andshowedthatitcanachieveaconstantper-nodethroughput.TheschemeovercomesthethroughputboundofO(1=p nlogn)1originallyestablishedbyGuptaandKumar[ 60 ]forastaticwirelessnetwork,wherenisthenumberofnodes.Althoughheavyuseofrelayingthroughnodemobilityallowsforhigherthroughput,italsobearstwonegativeside-effects:increaseddelayandincreasedstoragerequirements.Ithasbeenshownin[ 48 120 ]thatthe2-hoprelayingschemein[ 58 ]yieldsanextremelylargeaveragedelayof(n),whereastherelaybuffersizerequirementoneachnodeisatleast(n)[ 66 ].Sinceboththroughputanddelayareimportantnetworkperformancemetricsfromtheperspectiveofanapplication,signicanteffortinthelastfewyearshasbeendevotedtounderstandthethroughput-delayrelationshipinMANETs(refertoSection 4.2.1 andthereferencestherein)inthenetworkingresearchcommunity.AninterestingworkbyNeelyandModiano[ 120 ]suggestedtoutilizeredundantpacketstransmissionthroughmultipleopportunisticpaths(whicharecomposedofmultipleopportunisticlinks)ofaMANETtobalancetheconictingrequirementsonthroughputanddelay.Thebasicidea 1RefertoAppendixB forthestandardasymptoticnotationusedthroughouttheDissertation. 89

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isthatthetimerequiredforapackettoreachthedestination(i.e.,end-to-enddelay)canbereducedbyrepeatedlytransmittingthispackettomanyrelaynodesofthenetwork,andthusimprovingthechancesthatsomeuserholdinganoriginalorduplicateversionofthepacketreachesthedestinationnode.Clearly,thecostofthisapproachisthedecreasedthroughputsinceduplicatepacketswastescarceopportunitiesofwirelesstransmissions.Inparticular,withi.i.d.mobility,itwasshownthatforper-nodethroughputT(n)=O(1),therelayingstrategieswithreplicationcouldyieldend-to-enddelayD(n)scalingas(nT(n))[ 120 ].BufferspaceofmobilenodesinMANETsisalsoanimportantandscarcenetworkresource.Constraintsonbufferspace/storagereducethethroughputcapacityandincreasethenetworkdelayduetobufferoveroworpacketlosses.Inpractice,bufferspaceisalwayslimited,anditseffectsonnetworkperformanceshouldbequantied.In[ 66 ],HerdtnerandChongshowedthat,giventhesizeoftherelaybufferpernode,saybn,theper-nodethroughputisatmostO(p bn=n).Therelationshipofdelaywithstorageandtheimpactofreplicationstrategiesasproposedin[ 120 ],however,werenotaddressed.PreviousstudiesonthescalinglawsofMANETs,asdiscussedabove,areallbasedontheimplicitassumptionthateachnodecanonlyperformtraditionaloperationsonpackets,suchasstorage,replicationandforwarding.Recently,networkcoding,rstintroducedbyAhlswedeetal.[ 2 ]in2000,hasbeenwidelyrecognizedasapromisingprimitiveoperationbesidessimplereplicatingandforwardingincomingpackets[ 42 ].Usingtheparadigmofnetworkcoding,whenanodeisscheduledtotransmit,itmaytransmitamixedpacketasaresultofalgebraicoperationsonitsincomingpacketstomaximizetheusefulnessofthistransmissiontoallreceiversinitstransmissionrange.Moreover,whenanodereceivesanewpacketanditsbufferisfull,itwillmixthenewpacketwithstoredonesinsuchawaythatmaximizestheinformationstoredinitsbuffer.ItisworthnotingthataparticularusefulformofnetworkcodingcalledRandomLinear 90

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Coding(RLC)wasproposedintheliterature[ 67 73 ]toindependentlyandrandomlymixincomingpacketsateachnodewithlinearoperations,whichallowsthenodesofthenetworktoachievetheoptimalperformancewhileoperatinginadecentralizedfashion.Intuitively,whenRLCinsteadofreplicationisusedtominimizetheend-to-enddelay,networkcongestioncanbealleviatedandtherequirementonbuffersizecanberelaxed.Therefore,abetterthroughput-delay-storagetradeoffisexpectedtoobtained.SincenetworkcodingwasnottakenintoconsiderationinGrossglauserandTse'soriginalwork[ 58 ]andtherelatedwork[ 48 66 120 ]thatfollowed,aninterestingquestionraisednaturallyishowmuchbenetnetworkcodingcanprovidetothenetworkperformanceofMANETscomparedtowhenonlysimplereplicationandforwardingareallowedforrelaynodes.AnsweringthisquestionwillhelpusbetterunderstandnotonlythebenetsandlimitationsofnetworkcodinginwirelessnetworksbutalsothefundamentaltradeoffsdeterminingMANET'sperformance.InthischapterwestudythescalinglawsgoverningMANETs.Wecharacterizethethroughput-delay-storagetradeoffswithrespecttodifferentnodemobilitypatterns.Weidentifyscenariosinwhichnetworkcodingcanprovidesignicantimprovementonnetworkperformance.Wealsoprovideinsightsonwhenandhowinformationmixingisbenecialandproposealgorithmstoshowthatthesebenetscanbeachievedinaneffectiveanddecentralizedfashion.Therestofthechapterisoutlinedasfollows.Section 4.2 providesareviewofrelatedwork.Section 4.3 presentsmodelsofMANETsanddenitionsofnetworkperformancemetrics.Section 4.4 isincludedforcomparisonpurposes,whichsummariesmainresultsonthroughput-delay-storagetradeoffsforMANETsusingreplicationstrategies,insteadofnetworkcoding.Section 4.5 andSection 4.6 investigatethroughput-delay-storagetradeoffswhennetworkcodingisused.Finally,Section 4.7 concludesthechapter. 91

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4.2BackgroundandRelatedWorkThereexisttwomainbodiesoftheoreticalworkrelatedtothetopicaddressedinthischapter:thescalinglawsinMANETswherenetworkcodingisnotapplied,andthenetworkcodingapplicationsinMANETswherethescalinglawisnotstudiedintheoreticalanalysis.Wereviewbothveinsofworkandpresentrecentresultsonscalinglawsofthroughputinstaticwirelessnetworkswithnetworkcodingascomparison.Thecontributionsofourworkarealsodiscussed. 4.2.1ScalingLawsofMANETswithoutNetworkCodingSeminalworkofGuptaandKumar[ 60 ]initiatedtheinvestigationonhowthethroughputofwirelessnetworksscaleswithn,thenumberofnodes.Undertheassumptionthatnodeswithcommontransmissionrangearerandomlydistributed,itisshownthatper-nodethroughputforstaticwirelessnetworksscalesas(1=p nlogn).Notethat[ 60 ]implicitlyusedauidmodelforestablishingthroughputscaling.LaterworkbyKulkarniandViswannath[ 90 ]consolidatedtheresultof[ 60 ]withanexplicitconstantpacketsizemodel.In[ 46 ],withpercolationtheory,Franceschettietal.showedthatthe(1=p n)per-nodethroughputisachievableifeachnodecanadjustitstransmissionrange(throughpowercontrol),however,thethroughputvanishingproblemforlarge-scale(n!1)staticwirelessnetworksstillremains.In[ 58 ],GrossglauserandTseshowedthatthemobilityofthenodesinaMANETcanbeexploitedtoovercomethisproblem.The2-hoprelayingschemetheyproposedachievesaconstantper-nodethroughputatthecostofalargedelayontheorderofn[ 48 120 ].ThisresultrevealsthepossibilityoftradinglargerdelayforhigherthroughputorlowerthroughputforsmallerdelayinMANETs.Sincethen,aurryofresearchactivitieshavetriedtocharacterizethethroughput-delayrelationshipwithrespecttonodemobility,e.g.,[ 11 48 50 100 101 120 140 154 165 ].Ingeneral,therearetwowaystotradethroughputfordelayintheliterature.KleinrockandSilvester[ 87 ]maybethersttondthatdelayofmulti-hoprouting 92

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canbereducedbyincreasingthetransmissionradiusofeachrelaynode,attheexpenseofreducingthenumberofsimultaneoustransmissionsthenetworkcansupport,whichleadstoalowerthroughput.Similartransmissionradiusscalingtechniqueshaveappearedin[ 48 50 100 101 140 154 165 ].Anotherapproach,whichimprovesdelayviaredundantpackettransfersisconsideredin[ 120 146 ].Inthischapter,wefollowthisapproach,adoptingreplicationstrategyandcomparingitwithnetworkcodingforthefollowingreasons: Firstofall,theassumptionthattransmissionrangescanscalewithn,thenumberofnodes,isimpracticalforlarge-scaleMANETs.ToobtainthescalinglawofMANETs,weusuallyrequirentendingtoinnity,whichisequivalenttoassumingp An!1forextendednetworkmodel,whereAnistheareaofthenetwork(cf.Section 7.3.1 ).Ingeneral,wirelessdeviceispowerlimited,renderingitimpossibletorequirethetransmissionrangereachingtheorderofp An. Second,tradeoffstheoreticallyanalyzedusingtherstmeansmentionedabovearemainlybasedonuidmodel,inwhichthepacketsareallowedtobearbitrarilysmallasn!1(e.g.,[ 48 49 100 101 140 154 165 ]).Ontheotherhand,tradeoffsobtainedthroughthesecondapproachassumeconstantpacketsizemodel,wherethepacketsizeremainsconstant,i.e.,doesnotscaledownwithn(e.g.,[ 120 ]).Weprefertheconstantpacketsizemodelsinceinreality,packetsizedoesnotchangewhenmorenodesareaddedtothenetwork.Furthermore,uidmodelcannotbeappliedtoscenarioswithnetworkcoding,sinceeverycodedpacketincludesacodevectorofatleastconstantsize(cf.Section 4.5.1 )forsuccessfuldecoding. Secondly,inthischapterweareinterestedinexaminingpuregainsintroducedbynetworkcodinginMANETs.Replicationstrategiescanbereplacedbynetworkcoding,whichprovidesagoodchanceforcomparison.Transmissionradiusscalingtechniques,however,areorthogonaltonetworkcoding,andshouldbestudiedseparately.Bufferspace(storage)isanotherimportantnetworkresource,anditsimpactonthethroughputscalingwasinvestigatedin[ 76 ]and[ 66 ].Forstaticwirelessnetworks,Jelenkovic,MomcilovicandSquillante[ 76 ]showedthattherewasnoprotocolcapableofcarryingoutthelimitingper-nodethroughputof(1=p n)withnodesthathaveconstantbufferspace.Ontheotherhand,theyestablishedtheexistenceofaprotocolwhichrealizesthethroughputof(1=p nlogn)whennodeshaveconstantbufferspace.For 93

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mobilewirelessnetworks,HerdtnerandChong[ 66 ]establishedthatifrelaybufferspacesareboundedabovebyaconstant,mobilitydoesnotsubstantiallyincreasethethroughputofMANETs.Inparticular,theyshowedthatthethroughputisatmostO(p bn=n)pernode,wherebnisthesizeofrelaybufferspaces2.Asaconsequence,thethroughputofmobilenetworkswithnitebufferspacesisatmostO(1=p n)pernode,whichisalargeperformancedegradationcomparedtothecasewithinnitebufferspaceswhere(1)throughputisachievableasestablishedin[ 58 ].Notethat,issuesrelatedtobufferconstraintscanonlybeaddressedwithconstantpacketsizemodel.Foruidmodel,buffersarenotrequired[ 50 ]andrelateddiscussionsaremeaningless. 4.2.2NetworkCodingApplicationsinWirelessNetworksTheideathat,whenRLCisallowedinintermediatenodes,comparedtoreplicationstrategies[ 120 146 ],largerthroughputcanbeachievedwiththesamedelayandsmallersizesofnodebuffers,wasperhapsrstexplicitlydevelopedinthework[ 175 ]byZhangetal.,whereasimulation-basedstudyofthebenetofRLCinoneunicastcommunicationisalsopresented.TherecentworkbyLinandLi[ 102 ]givesarigorousanalysisofthisideabasedonordinarydifferentialequations.Toourknowledge,[ 175 ]and[ 102 ]aretheclosesttoourworkintermsofunderstandingtherelationshipsbetweenthroughput,delayandstoragewithnetworkcoding.However,ourworkhasthefollowingadvantages: Firstofall,insteadofexplicitlymodelingnodes'spatialdistributionsasinthischapter,themobilityofnodesin[ 175 ]and[ 102 ]ismodeledwithmeetingtimesofanypairofnodes,tosimplifytheanalysis.Theproblemisthat,themostimportantfeatureofwirelesstransmission,i.e.,interference,isnotincludedintheirmodeling.Itisneverthelessstillreasonablefor[ 175 ]and[ 102 ],sincetheauthorsaremainlyinterestedindelaytolerantnetworks,wherenodesareassumedtobesparselydistributedandinterferencefromsimultaneoustransmissionscanbeignored.However,itisobviouslynotsuitableforthestudyofgeneralMANETs. 2Wewillshowthatthisboundisnottight,i.e.,itcannotbeachievedbyanyrealscheme,inSection 4.4.2 94

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Second,thetrafcpatternconsideredinourchapterismorepractical.Thenumberofunicastsessionssupportedinthischapteris(n),whileonlyoneunicastorbroadcastsessionisassumedin[ 175 ]and[ 102 ]. Next,onlyepidemicroutinganditsreplacementofnetworkcodingareconsideredin[ 175 ]and[ 102 ],whileinourwork,severalalternativesareconsideredanddifferentalgorithmsaredevelopedwhichachievethroughput-delay-storagetriplesondifferentordersofn.Therefore,weobtainacompletecharacterizationoftradeoffsinMANETs. Mostimportantly,explicitexpressionsofnetworkperformanceortradeoffsareobtainedinourchapterforthersttime,whichprovideinsightsonthedegreeofscalabilityofMANETswithnetworkcoding.Thisisthebenetbroughtbyscalinglawbasedapproachadoptedinthischapter. 4.2.3ScalingLawsofWirelessNetworkswithNetworkCodingScalinglawsgoverningwirelessnetworkswithnetworkcodinghaveonlybeeninvestigatedinthelimitedscenariosintheliteraturerecently.Thedelaygainsandreliabilitybenet(measuredinthereducednumberoftransmissions)ofnetworkcodinginunreliablewirelessnetworkswerecharacterizedin[ 3 37 ]and[ 52 ],respectively.However,theseresultsareforonemulticastsessionwithone-hoptransmissionorstablenetworktopology.Formultipleunicastscenario,Liuetal.[ 105 106 ]andKeshavarz-Haddadtetal.[ 81 ]showedthatforstaticwirelessnetworks,networkcodingandbroadcastingatmostprovideaconstant-factoredimprovementinthethroughput,comparedtoGuptaandKumar's(1=p nlogn)per-nodethroughput[ 60 ].Howeverinthischapter,ourresultsshowthat,networkcodingcanprovidesignicantimprovementonnetworkperformancewhenmobilityisutilized,whichisimpossibleinstaticwirelessnetworks[ 81 105 106 ].WebelieveitrevealstheintrinsicdifferencebetweenMANETsandstaticwirelessnetworks.Veryrecently,Yingetal.[ 165 ]proposedjointcoding-schedulingschemestoimprovethethroughput-delaytradeoffsforMANETsusingrate-lesscodes(e.g.Raptorcodes).Notethattheirwork[ 165 ]isdifferentfromoursinthat:(1)Theyalsoconsideredadjustingtransmissionrange(orcellsize)toachieveabettertradeoff.Thereforeitis 95

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difculttosaythatthegainintheirschemeispurelyduetothecoding.(2)Theresultsin[20]arerestrictedtocaseswhendelayD(n)isboth!(3p n)ando(n).Inthischapter,wedesigntheschemeswithD(n)=(logn)andD(n)=(n). 4.3MANETModelsandDenitionsInthissection,werstpresentthemobilerandomnetworkmodelsalongwiththemodelforsuccessfulwirelesstransmissionusedinthischapter.Thedenitionsofnetworkperformancemetricssuchasthroughputanddelayarealsoprovided. 4.3.1NetworkModelsRandomnetworkmodelforMANETs:ConsideranadhocnetworkwherennodesareinitiallyuniformlydistributedatrandominasquaretorusofareaAn.Weconsideramultiple(n)unicastscenarioinwhicheachnodei2f1,2,,ngisasourcenodeforoneunicastsession,andadestinationnodeforanotherunicastsession.Supposethatthesourcenodeihasdataintendedfordestinationnoded(i).Weassumethateachsourcenodehasaninnitestreamofpacketstosendtoitsdestination.Thesource-destination(S-D)associationdoesnotchangewithtime,althoughthenodesthemselvesmove.Mobilitymodels:Thetorusisdividedintom=(n)squarecellsofareaAn=meach,resultinginatwo-dimensionalp mp mdiscretetorus,seeFigure 4-1 foranillustration.Theinitialpositionofeachnodeisequallylikelytobeanyofthempossiblecellsindependentofothers.Wefurtherassumethetimeisslottedandwestudythefollowingmobilitymodelsinthischapter: Two-dimensionali.i.d.mobilitymodel(fastmobilitymodel):Ateachtimeslot,nodesrandomlychooseanewcelllocationindependentlyandidentically(i.i.d.)distributedoverallcellsinthenetwork.Thismodelcapturesthesituationwhenmobileusermovessoquicklythatitspositionisalmostindependentfromtimetotime.Withthisassumption,thenetworktopologydramaticallychangesineverytimeslot,sothatthenetworkbehaviorcannotbepredictedandxedroutingalgorithmscannotbeused.Thismobilitymodelisalsousedin[ 101 120 140 154 165 ]. Two-dimensionalrandomwalkmodel(slowmobilitymodel):Letanodebeincell(i,j)2f1,p mg2attimeslott,then,attimeslott+1,thenodeisequally 96

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Figure4-1. FastandslowmobilitymodelsforMANETs. likelytobeinthesamecell(i,j)oranyofthefouradjacentcellsf(i)]TJ /F6 11.955 Tf 12.5 0 Td[(1,j),(i+1,j),(i,j)]TJ /F6 11.955 Tf 12.1 0 Td[(1),(i,j+1)g,whereadditionandsubtractionaremodulop m.Soeachnodeinfactindependentlyperformsasimplerandomwalkonthetwo-dimensionalp mp mdiscretetorus.Notethatthismodelimplicitlysetsanupper-boundonthevelocityofmobilenodesasp 2An=m.Therefore,itisasuitablemodelforcapturingrealmotionofnodeswithslowmobility.Similarmobilitymodelisalsoadoptedin[ 48 50 140 165 ].Modelforsuccessfultransmission:Forcharacterizingtheconditionforasuccessfultransmission,weadopttheProtocolmodelasdenedin[ 60 ].Weassumethatallnodesuseacommonrangercfortheirtransmissions,andatransmissionfromnodeitonodejissuccessfulifandonlyifdijrcanddkj(1+)rcforanyothersimultaneoustransmitter,saynodek.Here,dijisthedistancebetweennodesiandj,andisapositiveconstantindependentofn.Duringasuccessfultransmission,nodessenddataataconstantrateofWbitspersecond.Concurrentlytransmittingcells:Nowwedenethetransmissionrangeandschedule.Wechooserninsuchawaythatanynodeinallcellscanalwaysdirectlytransmittoanyothernodeinthesamecellusingthesmallestcommonrangeoftransmission.Obviously,rc=p 2sn=p 2An=m=(p An=n). 97

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Timeisslottedforpacketizedtransmission.WeassumeonlyO(1)packetscanbetransmittedpercellpertimeslot,i.e.,ouranalysisisexplicitlybasedonconstantpacketsizemodel(referto[ 50 ]andSection 4.2.1 fordetaileddiscussions).Wesaythatacellisactiveinatimeslotifanyofitsnodestransmitsinthattimeslot,andacell(i,j)interfereswithanothercell(k,l)ifatransmissionbyanodeincell(i,j)canaffectthesuccessofasimultaneoustransmissionbyanodeincell(k,l).ConsiderinProtocolmodel,twointerference-freecellsverticallyorhorizontallyK)]TJ /F6 11.955 Tf 12.21 0 Td[(1cellsapart.Weknowthatinordertoguaranteethesuccessfultransmissionsinthesetwocells,weneed(K)]TJ /F6 11.955 Tf 12.4 0 Td[(1)sn(1+)rn(refertonodes'positionsatthetoprightofFigure 4-2 foranillustration).LetK=1+(1+)p 2,wedivideallcellsintoK2groups.AllcellsbelongingtothesamegroupareatverticalorhorizontaldistanceofexactlysomemultiplesofK,andcantransmitsimultaneouslyasdepictedinFigure 4-2 .Now,wecandesignanitelengthtime-divisionschedulingschemeofK2timeslots,inwhicheachcellgroupisassignedoneslottotransmit.Figure 4-2 givesanexampleofsuchcellschedulingwithK=4.Basedontheabovediscussion,wehavethefollowingProposition. Proposition4.1. UndertheProtocolmodel,thereexistsaninterference-freeschedulesuchthateachcellbecomesactiveregularlyonceinK2timeslotsanditdoesnotinterferewithanyothersimultaneouslytransmittingcells.HereKdependsonlyon,andisindependentofn.Extendednetworkmodel:WeareparticularlyinterestedinasymptoticpropertiesofMANETs,whichholdwithhighprobability3forlarge-scaleMANETs.Therefore,weneedoftentakelimitsasn!1.WhentheregionareaAnisxed,itcorrespondstoadensenetworkmodel[ 60 130 ],sincethedensityofthenetworkd=n=Analsotends 3Wesaythataneventoccurswithhighprobability(w.h.p.)ifitsprobabilitytendsto1asn!1. 98

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Figure4-2. Celltransmissionscheduling.HereisanillustrationofthecellsbeingdividedintoK2groupsforthecaseofK=4,i.e.,16groups.Allthebluecellswhichareingroup1transmitinthesametimeslot.Inthenexttimeslotallthecellsingroup2transmitandsoon. toinnityasn.Anotherwidelyusedmodelistheextendednetworkmodel[ 34 114 ],inwhichboththenumberofnodesandtheareaoftheregionAngotoinnitywhilediskeptconstant.Bothmodelsarewidelyusedintheliteratureandwewillfocusonthelatterone.Intheextendednetworkmodel,An=n=d=(n),andcorrespondinglyrc=(p An=n)=(1),whichisindependentofn.Thisismorepractical,sincepowerconstraintonwirelessdevicesdoesnotchangewhenmorenodesareaddedtothenetwork.Wenotethat,however,resultsobtainedinthischaptercanbeeasilyextendedtodensenetworkmodel. 4.3.2NetworkPerformanceMetricsDenitionofthroughput:Athroughput>0issaidtobefeasible/achievableifeverynodecansendatarateofbitspersecondtoitschosendestination.WedenotebyT(n),themaximumfeasiblethroughputw.h.p.Givenascheme,letM(i,t)bethenumberofpacketsfromsourcenodeithatdestinationnoded(i)receivesinttimeslotsunderscheme,for1in.Notethatthiscouldbearandomquantityforagivenrealizationofthenetwork.DenethelongtermthroughputofS-Dpairi,denotedby 99

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i(n),tobei(n)=liminft!11 tM(i,t).SchemeissaidtohavethroughputT(n)iflimn!1P)]TJ /F7 11.955 Tf 5.47 -9.68 Td[(i(n)T(n)foralli=1.Weallowrandomnessinschemesandasaresultrandomquantitiesaboveareinthejointprobabilityspaceincludingboththerandomnetworkofsizenandthescheme.Notethatwhennetworkcodingisutilizedinscheme,M(i,t)isthenumberofsuccessfullydecodedpacketsreceivedbythedestinationd(i)ofS-Dpairiinttimeslotsunderscheme.Denitionofdelay:Thedelayofapacketisthetimeittakesthepackettoreachthedestinationafteritleavesthesource.Wedonottakequeueingdelayatthesourceintoaccount,sinceourinterestisinthenetworkdelay.LetDi(j)denotethedelayofpacketjofS-Dpairiunderscheme,thenthesamplemeanofdelay(overpacketsthatreachtheirdestinations)forS-Dpairiis Di=limsupk!11 kkXj=1Di(j).TheaveragedelayoverallS-Dpairsforaparticularrealizationoftherandomnetworkisthen D=1 nPni=1 Di.ThedelayforaschemeistheexpectationoftheaveragedelayoverallS-Dpairsandallrandomnetworkcongurations,i.e.,D(n)=E D=1 nnXi=1Eh Dii.Notethatwhennetworkcodingisutilized,weconsiderthedelayofgettingoriginalpackets.WhenanoriginalpacketmibelongstothegenerationM,thedelayofmiunderschemeisthetimefromtherstpacketbelongingtoMdepartsthesourcetotheoriginalpacketmihasbeendecodedatthedestination. 100

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4.4Throughput-Delay-StorageTradeoffswithoutNetworkCodingInthissection,wegiveabriefoverviewoftheredundancy-basedschemesaspresentedin[ 120 ]andestablishthethroughput-delay-storagetradeoffsinMANETswithoutnetworkcoding. 4.4.1Throughput-DelayTradeoffswithInniteBufferSpacesWerstdescribethreerelayschemeswithdifferentredundancyproposedin[ 120 ]fromauniedpointofview.ThreeRedundancy-BasedSchemesProposedin[ 120 ]Wecancontrolthetransmissionredundancyofeachpacketwithtwomethods:thenumberofhopseachpacketwilltakefromsourcetodestination,andthetotalnumberofcopies(replicas)ofeachoriginalpacketinthenetwork.Thethreeschemes,namely,2-hoprelaywithoutreplicas,2-hoprelaywithk1replicas,andmulti-hoprelaywithk2replicasrepresentdifferentcombinationsofthetwomethods.Eachschemehastwoparts:(1)schedulingofactivecells;(2)schedulingoftransmissioninanactivecell.Thethreeschemeshavethesamecellschedulingpolicy(Part(1))asfollows: EachcellbecomesactiveonceineveryK2timeslotsasdiscussedinProposition 7.1 Inanactivecell,transmissionisalwaysbetweentwonodeswithinthesamecell.Ineveryactivecellwithatleasttwonodes,intra-celltransmissionscheduling(Part(2))isneeded. For2-hoprelayschemes,eachpacketatmosttakestwohopsfromsourcetodestination.Ateverytimeslot,atransmitter-receiver(T-R)pairisselectedrandomlyfromallnodepairsineachactivecell.Withprobability1=2,thechosenT-Rpairwillactinsource-to-relay(S!R)modeorrelay-to-destination(R!D)mode.Thedifferenceisthat,for2-hoprelaywithoutreplicas,packetsarenotduplicatedandareheldbyatmostonenode(sourceorrelay)atanytimeslot,whilefor2-hoprelaywithk1replicas,inS!Rmode,thesourcewillsendk1replicastodistinctnodesasrelays,andinR!Dmode,thereceivertellsitstransmitterwhichpacketitislookingforbeforethetransmissionbegins(usinghandshake),toguaranteethatthereceiveralwaysgetsnewusefulpacketswhenitactsasadestinationnode. 101

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Table4-1. Networkperformancesunderfastmobilitymodel SchemeThroughputDelay 2-hoprelaywithoutreplicas(1)(n)2-hoprelaywithk1replicas(1=p n)(p n)Multi-hoprelaywithk2replicas1 nlogn(logn) Multi-hoprelaywithk2replicasisjustanothertypeofoodingscheme,whichtransmitsk2replicasofeachpacket,andplacesnoconstrainsonthenumberofhops.Itassumesthateachpacketisstampedwiththetimeslottinwhichitrstleavesthesource.Ateverytimeslotineachactivecell,ofallpacketsthatarecontainedinatleastonenodeofthecellandthathaveneverbeenreceivedbyanyothernodeinthesamecell,thepacketthathasthesmallesttimestamptwillbeselectedtosendtoallnodesinthecell(i.e.,thelocaloldestpolicyforpackettransmissionscheduling). Theorem4.1. Assuminginnitebufferspaceateachnode,throughput-delaytradeoffsachievedbythethreeredundancy-basedschemesproposedin[ 120 ]forMANETsunderfastmobilitymodelcanbesummarizedinTable 4-1 .Notethattheperformanceaboveisachievedwithk1=(p n)andk2=(logn),respectively. Remark4.1. ThehandshakeusedinR!Dmodefor2-hoprelaywithk1redundancy,andthelocaloldestpolicyforpackettransmissionschedulingusedinmulti-hoprelaywithk2redundancy,areadoptedtoreduceunnecessarytransmissionredundancyanddelay,whichiscriticalforachievingoptimalthroughput-delaytradeoffs.NotethatthesetwotechniquescanalsobereplacedbyanaivetechniquecalledRandomMessageSelection(RMS)proposedin[ 33 ]and[ 43 ],whichselectsthepackettobetransmittedrandomlywithoutconsideringthereceiver'spreference.Therefore,itispossibleforRMStoschedulethetransmissionofapacketalreadyinthereceiver'sbuffer,causingdecreasedthroughputandincreaseddelay.Forexample,ithasbeenshownthatthedelayforRMSis(nlogn),muchlargerthanthatoftheschemesdescribedhere.ThebenetofRMSisthatitsimpliesthealgorithmrunningineachnode,andsavesthecommunicationoverheadintroducedbycomplextechniques.Forextendednetworkmodelusedinthischapter,nodedensityineachcellissmall,i.e.,d=(1).The 102

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Table4-2. Networkperformancesunderslowmobilitymodel SchemeThroughputDelay 2-hoprelaywithoutreplicas(1)(nlogn)2-hoprelaywithk1replicas1 p nlogn(p nlogn)Multi-hoprelaywithk2replicas1 np n(p n) overheadandcomplexityintroducedbythetworedundancy-basedschemescanbeignored.However,whenthenodedensityineachcellishigh,e.g.,indensenetworkmodel,optimalredundancy-basedschemesareimpractical,andRMSistheonlychoice,wherethegainofusingnetworkcodingwillbeampliedbyafactorof(logn)asshownin[ 33 ]and[ 43 ]. Theorem4.2. Assuminginnitebufferspaceateachnode,throughput-delaytradeoffsachievedbythethreeredundancy-basedschemesproposedin[ 120 ]forMANETsunderslowmobilitymodelcanbesummarizedinTable 4-2 .Notethattheperformanceaboveisachievedwithk1=(p nlogn)andk2=(p n),respectively. 4.4.2Throughput-StorageTradeoffsInthissubsection,weanalyzetheimpactsofniteorlimitedbuffersizeonthescalingpropertiesofMANETs.Notethat[ 66 ]onlyprovidesalooseupperboundonthroughputgivenbuffersizebn(cf.Section 4.2.1 ).Ourresultspresentedhereprovideatighterboundthanthepreviousonesin[ 66 ]. Theorem4.3. Assumeeachnodehasabufferspaceofbnpackets,thethroughputisupper-boundedbyO(bn bn+n),andthisboundisachievableusing2-hoprelaywithoutreplicas. Remark4.2. ForagivenS-Dpair,theaveragepacketqueuesizeinthenetworkistheproductofpernodethroughputandaveragepacketlifetimeusingLittleslaw.Sincethepacketlifetimeisequaltotheaveragedelay,theaveragenumberofpacketsforagivenpair(i.e.,thenumberofbuffersrequiredtosupportagivenS-Dpair'sow)isdeterminedbythethroughput-delayproduct.Fromthisobservation,theaveragebuffer 103

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requirementsforsupportingoptimalthroughput-delaytradeoffsinTheorem 4.1 areO(n),O(1),andO(1=n),respectively.Therefore,wecanconcludethatthehighertheredundancyinascheme,thelowertherequirementonthebuffersize.Atarstglance,thisresultconictswithourintuitionsinceexcessivereplicationwilltendtowastebufferspace.However,theresultspresentedinSection 4.4.1 showthatreplicationreducesboththroughputanddelaybecausewetradethroughputfordelayimprovement.Asaresult,thethroughput-delayproductdecreasesaswell. 4.5Throughput-Delay-StorageTradeoffswithNetworkCoding:SchemesandResultsWerstreviewRLCusedinournetworkcodingbasedschemes.Thisbearsexactlythesamesetupasin[ 33 ].ThenwedescribetheschemesdevelopedforanalyzingtradeoffsinMANETswithnetworkcoding,andidentifyscenariosinwhichRLCimprovesnetworkperformanceofMANETssignicantly. 4.5.1NetworkCodingOperationRandomlinearcoding(RLC)isappliedtoanitesetofkoriginalpackets(i.e.,M=fm1,m2,,mkg),thatiscalledageneration.Eachpacketisviewedasanr-dimensionalvectoroveraniteeld,Fqofsizeq,i.e.,mi2Frq,i=1,2,,k.Ifthepacketsizeismbits,thiscanbedonebyviewingeachpacketasanr=dm=log2(q)e-dimensionalvectoroverFq(insteadofviewingeachpacketasanm-dimensionalvectoroverbinaryeld).Typically,F28(i.e.,F256)isused.AlltheadditionsandmultiplicationsinthefollowingdescriptionareassumedtobeoverFq.WeassumethatallthekpacketsinMarelinearlyindependent.DuringtheexecutionofaRLCbasedrelayscheme,thedestinationnodeofMcollectslinearcombinationsofthepacketsinM.Oncetherearekindependentlinearcombinationsatanode,itcanrecoveralltheoriginalpacketsinMsuccessfully.Now,consideracertaintimeslott.LetSv(t)andSu(t)denotethesetofallthecodedpackets(eachcodedpacketisalinearcombinationofthepacketsinM)atnode 104

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vandu,respectively,atthebeginningofthetimeslott.Moreprecisely,ifcodedpacketfl2Sv(t),wherel=1,2,,jSv(t)j,thenfl2Frqhastheformfl=Pki=1limi,li2Fq.Theschemeensuresthatali'sareknowntonodevbyappendingeachpacketflwithacodevector,whichwillbeexplainedalittlelater.LetSv(t))]TJ /F1 11.955 Tf 10.4 -4.34 Td[(andSu(t))]TJ /F1 11.955 Tf -456.78 -28.25 Td[(denotethesubspacesspannedbythecodedpacketsinSv(t)andSu(t),respectively.IfSv(t))]TJ /F12 11.955 Tf 11.09 -4.33 Td[(*Su(t))]TJ /F1 11.955 Tf 7.09 -4.33 Td[(,wesaynodevhasusefulinformationaboutMforu.Intimeslott,ifnodevisscheduledbytheschemetotransmitapacketrelatedtoMtonodeu,vrstchecksifithasusefulinformationforu.Ifso,vtransmitsarandomcodedpacketwithpayloadfnew2Frqtou,wherefnew=Xfl2Sv(t)lfl,l2FqandP(l=)=1 q,82Fq.Itiseasytocheckthatfnewisstillalinearcombinationofthekoriginalpackets,andcanbewrittenasfnew=Pki=1imiwherei=Pfl2Sv(t)lli2Fq.Fordecodingpurposes,thevector(1,2,,k)2Frq,calledcodevector,willbeappendedtofnew,andsentasoverhead.Thisoverheadclearlyrequiresapaddingofadditionalklog2(q)bits.Ifthepacketsizemlog2(q),whichwouldbethecaseunderourconstantpacketsizemodel,thentheoverheadrequiredbytheRLCbasedschemecanbeignoredinouranalysis.4Wesaythatvsendsaninnovativecodedpacketfnewtou,iffnewcanincreasethedimensionofthesubspaceSu(t))]TJ /F1 11.955 Tf 7.09 -4.34 Td[(,i.e.,dim(Su(t))]TJ /F6 11.955 Tf 7.08 -4.34 Td[().Notethatdim(Su(t))]TJ /F6 11.955 Tf 7.09 -4.34 Td[()kingeneralandifdim(Su(t))]TJ /F6 11.955 Tf 7.09 -4.34 Td[()=k,nodeucanrecoverallthekoriginalpacketsatonce.WenowrecallthefollowingkeyresultaboutRLC,whichsaysthatfnewwillbeaninnovativecodedpacketforuwithprobabilitynolessthan1)]TJ /F9 7.97 Tf 13.32 4.71 Td[(1 q. 4Moreprecisely,theconstantpacketsizemodelfororiginalpacketsmeansthatthepacketsizescalesas(logn)bits,sinceitneedstocarrytheIDofthedestinationnodewith(logn)bits.Forafaircomparison,werequirethatk=O(logn)forthecodedpacketsthroughoutthechapter.ThereforetheoverheadintroducedbythecodevectorwillnotchangetheorderofourresultsonT(n)andD(n)forRLC-basedschemes. 105

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Proposition4.2. (Lemma2.1in[ 33 ])LetSu(t)+=Su(t)[ffnewgbethesubspacespannedbythecodevectorsinuattheendoftimeslott,i.e.,afterreceivingacodedpacketfnewfromvaccordingtotheRLCbasedschemedescribedasabove.Then,P)]TJ /F22 11.955 Tf 5.48 -9.68 Td[(dim(Su(t)+)>dim(Su(t))]TJ /F6 11.955 Tf 7.08 -4.93 Td[()jSv(t))]TJ /F12 11.955 Tf 10.4 -4.93 Td[(*Su(t))]TJ /F15 11.955 Tf 7.09 4.75 Td[(1)]TJ /F6 11.955 Tf 13.4 8.08 Td[(1 q. 4.5.2RLC-BasedRelaySchemesInthissubsection,wedescribeRLC-basedrelayschemeswithdifferentroutingstrategies,whichwillbeusedlatertoexploitthroughput-delaytradeoffsinMANETs.Werstintroducetheconceptofbiggeneration.Inwhatfollows,whenwesaythatthesourcenodegroupsk=!(logn)originalpacketsintoonebiggeneration,weinfactseparatethesekpacketsintok=(logn)generations,eachwith(logn)packets.Whenthedestinationnodetriestodecodeoneoriginalpacket,itrstneedstocollect(k)codedpacketsfromthebiggeneration(with(logn)codedpacketsfromeachgeneration).ThereforetheoverheadintroducedbyRLCisignorableinouranalysis(cf.footnote 4 ).Schemes1:2-hopRelaywithRLC(1)koriginalpacketsineachsourcenodewillbegroupedintoone(big)generation.Eachsourcewillsendm=(1+)kcodedpacketsforeach(big)generation,whereisaconstant.(2)Codedpacketsforeachgenerationwillhavethesametimestamptp.Thevalueoftpisthetimetherstcodedpacketofthatgenerationleavesthesource.Allcodedpacketsofagenerationwillbedeletedfromtherelaybufferatthetimeslottift)]TJ /F3 11.955 Tf 11.96 0 Td[(tp>thp,wherethethresholdthpdependsonD(n)oftheschemeandwillbesufcientlylargerthanD(n).(3)EachcellbecomesactiveonceineveryK2timeslotsasdiscussedinProposition 7.1 .Inanactivecell,transmissionisalwaysbetweennodeswithinthesamecell. 106

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(4)Foranactivecellwithatleasttwonodes,arandomtransmitter-receiverpairisselected,withuniformprobabilityoverallpossiblenodepairsinthecell.Withprobability1=2,thetransmitterisscheduledtooperateineitherSource-to-RelayorRelay-to-Destinationmode,describedasfollows: Source-to-RelayMode:Thetransmittersendsacodedpacketofitscurrentgeneration,anddoessouponeverytransmissionopportunitywhileitisinsource-to-relaymodeuntilmcodedpacketshavebeendeliveredtodistinctnodes.Ifallothernodesinthecellalreadyhaveonecodedpacketforthatgeneration,thesourcewillbegintotransmitcodedpacketsfromthenextgeneration.EverynodestoresasinglepacketperS-Dpairpergeneration.Whenthenodereceivesanewpacket,arelaylinearlycombinestheincomingpacketwiththestoredone,andreplacesthestoredpacketwiththeresult.Notethatthenodesoperateinbroadcastmode,i.e.,everynodewillheareverytransmissioninitsrange,andupdatethepacketstorageasdescribedabove. Relay-to-DestinationMode:Ifthedesignatedtransmitterhasacodedpacketinitsrelaybufferforthedestinationnode,andtherankofcodedpacketsofthatgenerationinthereceiverissmallerthank,thecodedpacketistransmittedtothedesignatedreceiver. Remark4.3. Sincem>k,weneedamechanismtostopunnecessaryrelayofcodedpacketsofagenerationwhenitisalreadydecodedinthedestination.Hereweuseaproactivestoppingmechanism,i.e.,thetimestampofeachgeneration,sincewecanboundthedelayofthescheme.Intheanalysispartpresentedlater,wewillshowthatk=(n),andD(n)forthisschemeisalso(n)forfastandslowmobilitymodels.Therefore,thpshouldbelargerthan(n).Morecomplicatedreactivestoppingmechanisms(cf.[ 102 ]andthereferencestherein)canbeadoptedtoenhancetheefciencyoftheschemeinpractice.However,wefollowthesimplestdesignforanalyticaltractabilityofthescheme.Schemes2:Multi-hopRelaywithRLC(1)koriginalpacketsineachsourcenodewillbegroupedintoone(big)generation.Eachsourcewillsendm=(1+)kcodedpacketsforeachgeneration,whereisaconstant.Twotimestampsforeachgenerationareused.Oneiscalledthegenerating 107

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timetg,basedonthetimeforkoriginalpacketstobegroupedintoagenerationinthesource.Anotheriscalledtransmissiontimetp,basedonthetimetherstcodedpacketofthatgenerationistransmittedbythesource.(2)EachcellbecomesactiveonceineveryK2timeslotsasdiscussedinProposition 7.1 .Inanactivecell,transmissionisalwaysbetweennodeswithinthesamecell.(3)Foranactivecellwithatleasttwonodes,performthefollowing:amongallpacketscontainedinatleastonenodeofthecellandwhichhaveusefulinformationforsomeothernodeinthesamecell,choosethepacketwiththesmallestgeneratingtimetg.Ifthereareties,choosethepacketfromtheS-Dpairiwhichmaximizes(tg+i)modn.Transmitthispackettoallothernodesinthecell.Iftheselectedpacketisinthesource,thenthesourcewilltransmitthelinearcombinationofitskoriginalpacketsofthesamegeneration,insteadofaparticularpacketbelongingtothatgeneration.(4)EverynodestoresasinglepacketperS-Dpairpergeneration.Whenthenodereceivesanewpacket,arelaylinearlycombinestheincomingpacketwiththestoredone,andreplacesthestoredpacketwiththeresult.(5)Allcodedpacketsofagenerationwillbedeletedfromtherelaybufferatthetimeslottift)]TJ /F3 11.955 Tf 12.09 0 Td[(tp>thp,wherethethresholdthpdependsonD(n)oftheschemeandshouldbesufcientlylargerthanD(n). Remark4.4. ThegeneratingtimestamptgisusedtoconstructaoodingschemeforoneparticularS-DpairwhereallnS-Dpairsareactiveandsharethenetworkresource.Itiseasytoseethatthepacketsfromtheoldestgenerationthathasnotbeendeliveredtoallnodeswilldominatethetransmissionsoverthewholenetworkveryquickly.Thelong-termfairnessbetweenallS-Dpairsisguaranteedsinceinthecaseofties,packetsfromS-Dpairiaregiventoppriorityineveryntimeslots.Alsonotethat,sinceatoneparticulartimeslot,onlyonegenerationfromoneS-Dpairdominatesthewholenetwork,thenumberofpacketseachrelayneedstostoreinstep(4)is1,i.e.,justforonegenerationw.h.p.Anothertimestamptpusedherehasthesamefunctionalityasthe 108

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previousscheme.ThethresholdthpshouldbelargerthanD(n),scalingas(logn)and(p n),respectively,forfastandslowmobilitymodels. 4.5.3MainResultsaboutRLC-BasedSchemesInthissubsection,wesummarizetheperformanceoftheaboveschemesunderdifferentmobilitymodels.Here,wefocusontheintuitionandexplanationoftheseresults.Proofsofthesesresultswillbegiveninthenextsection. Theorem4.4. When2-hoprelaywithRLCschemeisusedandk=(n),wehaveT(n)=(1)andD(n)=(n)forfastandslowmobilitymodels. Remark4.5. ComparetoTheorems 4.1 and 4.2 ,itiseasytoseethat,RLCprovidesdelayimprovement(logn)underslowmobilitymodel.Nogainisfoundunderfastmobilitymodel.Itisnotsurprising,since2-hoprelaywithRLCschemeisusedtore-place2-hoprelaywithoutreplicas,andweknowthatinthelatter,thereisnoduplicatedpacketsinordertomaximizethethroughput.Thuswecannotexpectanygainswhennetworkcodingisused.Thegain(logn)ofdelayunderslowmobilitymodelcomesfromthelowerinformationpropagationspeed,andthemixingofpacketsincreasethisspeedbyguaranteeingthateverypacketthedestinationreceivedfromrelaynodeswillcontributesomeinformationforthedecodingofthepacketfromthesamegeneration.Forfastmobilitymodel,thisbenetvanishessincetheinformationpropagationspeedishighenough,andthedelayforwaitingkcodedpacketsfordecodingdominatesthewholedelay. Theorem4.5. Whenmulti-hoprelaywithRLCschemeisused,underfastmobilitymodelwithk=(logn),wehaveT(n)=(1=n)andD(n)=(logn).Underslowmobilitymodelwithk=(p n),wehaveT(n)=(1=n)andD(n)=(p n). Remark4.6. Underfastandslowmobilitymodels,multi-hopRLC-basedschemesalwaysprovidesignicantgainscomparedtooodingschemes.WecanseethattheRLC-basedschemecanachieveminimaldelay,withanimproveddelay-constrained 109

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Figure4-3. TimetablesfordifferentRLC-basedschemesunderslowmobilitymodel. throughput.Theintuitionisthat,whenoodingisused,thereexistenoughopportunitiestoenhanceperformancebyreplacingreplicaswithmoreintelligentcoding.Figure 4-3 comparestimetablesof2-hopandmulti-hopRLC-basedrelayschemes.Itcanbefoundthatin2-hoprelayschemes,multiplesessionsoperateinaparallelfash-ion,whileinmulti-hoprelayschemes,theyoperateinasequentialfashion.Therefore,ateachtimeslot,for2-hoprelayschemes,trafcpatternisstillmultipleunicasts.Formultipleunicasts,weseldomndgainsfromnetworkcoding.Whileformulti-hoprelayschemes,ateachtimeslot,trafcpatternlooksmorelikeonebroadcastsession,wheregainsfromnetworkcodingarenaturallyexpected. Remark4.7. Alsonoticethat,multi-hoprelayschemescanbedividedintomultiplephases,andineachphase,relayingforonegenerationfromoneS-Dpairwilldomi-natethenetwork,whichisinfactatypeofinformationoodinginthisphase(refertoFigure 4-3 (b)foranillustration).Theresultisthatineachphase,packetsfromonegenerationwillbebroadcastedtothewholenetwork,andiftheothern)]TJ /F6 11.955 Tf 12.47 0 Td[(1nodesarereceivers,theycanalldecodetheoriginalpacketsinthatgenerationattheendofthatphase.Soitguaranteesthatmulti-hoprelaywithRLCcodingcansupportall-to-alltrafcpattern(nbroadcastsessions)withthesameperformance.Notethatthisalsomeansthatthesamenetworkperformancecanbeachievedforanynmulticastsessions(sincereceiversinthiscasearejustasubsetofreceiversinthebroadcastcase).From 110

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Theorem 4.5 ,wecaneasilyobtainthefollowingcorollaryontheperformanceofmultiplebroadcastsandmulticastswithnetworkcoding. Corollary4.1. Forall-to-allcommunicationsoranymulticastswithnsources,whenmulti-hoprelaywithRLCschemeisused,underfastmobilitymodelwithk=(logn),wehaveT(n)=(1=n)andD(n)=(logn).Underslowmobilitymodelwithk=(p n),wehaveT(n)=(1=n)andD(n)=(p n).In[ 43 ],Fragoulietal.designedanRLC-basedschemebasedonresultsfrom[ 33 ].Forall-to-allcommunications,theyshowedthattheirschemeachievesT(n)=(1=n)andD(n)=(n)underfastmobilitymodel.Obviously,theirschemeobtainedthesamethroughputasoursatthecostofmuchlargerdelay.Thebasicideaoftheirschemeisthat,kpacketsfromkdifferentsourceswillbegroupedintoonegeneration,andtherelayingschemeisessentiallythesameasours.ThecomparisonhereraisesaninterestingquestionwhyinourRLC-basedschemesweonlymixpacketsfromthesamesource?Thereasonsarethefollowing:rstofall,asshownintheabovecomparison,evenforall-to-allcommunicationscenarios,mixingpacketsfromdifferentsourcesisnotagoodchoice.Second,formultipleunicastscenarios,wemixpacketsfromdifferentsourcesandthesepacketshavedifferentdestinations.Whenonedestinationdecodesapacketdesignatedforanotherdestination,thispacketisinfactaduplicateattherstdestinationwhichwillreducethethroughput.Inourmulti-hoprelaywithRLC,wealsointroduceredundancyforthesamereason.However,theredundancyhereisexplicitlydesignedfordecreasingthedelay.Whilefortheformercase,itispurelyawasteofnetworkresourceinmultipleunicastscenarios.Finally,groupingpacketsfromdifferentsourcesrequirescoordinations.Wearenotsureaboutthecostforperformingthiscoordinationtask,andweareinterestedindesigningfullydecentralizedschemes,inwhichtheoperationsfromdifferentnodesshouldbedecoupledasmuchaspossible.Fromtheabovediscussion,wecanascertainthatfor2-hoprelaywithRLC,haveonepacketperS-Dpairinanyrelaynodeisenoughforobtainingtheperformance 111

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proposedinTheorem 4.4 ,whileonepacketpernodeisenoughformulti-hoprelaywithRLC.Wecanconcludethatthestoragerequirementsforthesetwoschemesareexactlynand1,respectively,foreachnode.Comparedtotheresultsonthestoragerequirementsforredundancy-basedschemesinTheorem 4.3 ,weobtainthefollowingCorollary: Corollary4.2. RLCcannotprovideimprovementonstoragerequirementsbetterthanaconstantfactor.Itiseasytocheckthatthethroughput-storagetradeoffsobtainedbyRLC-basedschemesinthischapterfollowthesameprincipleprovidedinTheorem 4.3 4.6Throughput-Delay-StorageTradeoffswithNetworkCoding:Analysis 4.6.1PreliminariesThefollowinglemmaisusefulindelayanalysis,sinceitconrmsthattheeffectof(intra-cell)transmissionschedulingonlycontributestoaconstantfactor,whichcanbeignoredinasymptoticanalysis.Therefore,thetimefortwodesirednodestomeetwilldominatethedelayofthescheme. Lemma3. Intheschemesmentionedabove,everynodewillbescheduledtotransmitorreceiveapacketineachtimeslotwithaconstant,non-vanishingprobabilitythatisindependentofn. Proof. ThisresultcanbeobtainedfromProposition 7.1 .Itonlydependsonthesteadystatenodelocationdistribution.Notethatfastandslowmobilitymodelshavethesamenodelocationdistributionsinthesteadystate.Therefore,thisresultappliestobothmobilitymodels. Tofacilitatethetheoreticalanalysis,weneedrstinvestigatetwocriticaldelaysforfastandslowmobilitymodels:minimaldelaysfor2-hoprelaysandforooding.Here,2-hoprelayrepresentsanyschemewithcontrolledredundancyonthenumberofhops(inthe2-hoprelaycase,thenumberofhopsforeachpacketis2,andotherschemes 112

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withconstanthopconstrainswillyieldthecriticaldelaysonthesameorderofn),andoodingrepresentsallschemesthatremovethisconstrainttotally.Considerthefollowingsituation:initially,onlyonenode'scolorisred,whichwecallthesource.Allothernodesareblue.Wheneverasourcenodemeetsabluenode5,thelatteriscoloredred.Thetimefor(n)nodestobecomerediscalledminimal2-hopdelay.Ifwechangetheruleslightly:wheneverarednodemeetsabluenode,thelatteriscoloredred,thenthecorrespondingtimeisnamedasminimaloodingdelay.Obviously,thesetwocriticaldelaysreecttheintrinsicpropertiesofhowmobilitywillfacilitateinformationpropagation.Thesetwoquantitiesarescheme-independent,i.e.,theyholdforanyschemewithorwithoutreplicasandwithorwithoutnetworkcoding.Werstconsidertwocriticaldelaysforfastmobilitymodel. Lemma4. Theminimal2-hopdelayandtheminimaloodingdelayunderfastmobilitymodelare(n)and(logn),respectively. Proof. (a)Fortheminimaloodingdelay,Lemma3in[ 120 ]alreadyestablishedthatthenumberoftimeslotsrequiredforthesourcenodetosendpacketstoatleastn=2nodesis(logn).(b)Fortheminimal2-hopdelay,weusethefollowingball-into-binargument:Foragivensourcenode,wehave(n)]TJ /F6 11.955 Tf 12.57 0 Td[(2)possiblerelaynodes(i.e.,(n)]TJ /F6 11.955 Tf 12.56 0 Td[(2)bins).Ateachtimeslot,thesourcenodeselect1relaynodeandsendadistinctcodedpackettoit(i.e.,droppingoneballinaselectedbin)withaconstantprobability(cf.Lemma 3 ).Notethatweignorethisprobabilityinthefollowinganalysis,sinceitwillnotchangetheorderof 5Wesaythatnodeumeetsnodevifandonlyifnodesuandvareinthesamecellandscheduledasthesender-receiverpair.Recallthatforrandomwalkmobilitymodeldiscussedintheliterature,meetingisdenedastwonodesbeinginthesamecell.Inthischapterwedonotdistinguishthesetwokindsofmeetssinceinourmodelthetotalnumberofcellsis(n).Therefore,basedonLemma 3 ,the(intra-cell)transmissionschedulingonlycontributestoaconstantfactor,whichcanbeignoredinthefollowingasymptoticanalysis. 113

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ourresult.Repeatthism=(n)times(i.e.,mballsinto(n)]TJ /F6 11.955 Tf 12.82 0 Td[(2)bins),anddenotethenumberofdistinctrelaynodesasN(i.e.,thenumberofnon-emptybins).Here,weneedtoprovethatN=(n)forsomem="nwhere"isaconstant.Obviously,everybinisemptywithprobabilityexp()]TJ /F3 11.955 Tf 9.3 0 Td[(m=n)independentlywhenngoestoinnity.Weassumethesenbinsasap np ngrid.Basedonsitepercolationresults[ 114 ],when1)]TJ /F6 11.955 Tf 11.96 0 Td[(exp()]TJ /F3 11.955 Tf 9.3 0 Td[(m=n)0.60,thenumberofnon-emptybinsisontheorderof(n).Therefore,wecanalwaysndaconstant",suchthatN=(n)withm="n. Next,wepresenttheresultsforslowmobilitymodel. Lemma5. Theminimal2-hopdelayunderslowmobilitymodelis(n). Proof. Hereweneedtoshowthatafterm=(n)timeslots,thereareN=(n)distinctrednodesinthenetwork.BasedonLemma 3 ,thesourcenodeu(therstrednode)willbescheduledtotransmitm`=(n)timesinmtimeslots.ObviouslyNm`sincewecannotguaranteethateverytimewhennodeuisscheduledtobethesender,itwilltransmitapackettoanodeitnevermetbefore.LetN(u,v,m)denotethenumberoftimesthatnodeumeetsvwithinmtimeslots.Underrandomwalkmobilitymodel,thejointpositionoftwonodesduetoindependentrandomwalkscanbeviewedasadifferencerandomwalkrelativetothepositionofonenode.Thentheinter-meetingtimesarejusttheinter-visittimesofcell(1,1)forthedifferencerandomwalkonap np ntorus.Letbetherandomvariablerepresentingtheinter-meetingtimedenedasabove,ElGamaletal.[ 49 ]provethat: Lemma6. E[]=nandVar[]=(n2logn).Therefore,N(u,v,m)=m==(n=)(cf.footnote 5 ).RecallthatforarandomvariableX,wehaveVar[f(X)](f`(E[X]))2Var[X].Therefore,form=(n),wehaveE[N(u,v,m)]=(1)andVar[N(u,v,m)]=(logn).LetVbethesetofdistinctnodesthatsourcenodeumeetsinmtimeslots.Fortwodistinctnodesv1andv22V,N(u,v1,m)andN(u,v2,m)aretwoindependentrandom 114

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variableswiththesamedistribution.NotethatEPv2VN(u,v,m)=m.BecauseEPv2VN(u,v,m)=E[N]E[N(u,v,m)],weobtainthatE[N]=(n).FortworandomvariablesXandY,ifVar[X]exists,wehavethegeneralformulaforvariancedecompositionasthefollowing:Var[X]=Var[E[XjY]]+E[Var[XjY]].Therefore,Var"Xv2VN(u,v,m)#=E"Var"Xv2VN(u,v,m)N##+Var"E"Xv2VN(u,v,m)N##=E[N]Var[N(u,v,m)]+Var[N]E[N(u,v,m)]=(n)(logn)+Var[N](1).AlsonotethatjVjm,thereforeVar"Xv2VN(u,v,m)#mVar[N(u,v,m)]=(nlogn).FromabovediscussionsonVarPv2VN(u,v,m),weobtainthatVar[N]=O(nlogn).ByChebyshevinequality,forany0<<1,PfN(1)]TJ /F7 11.955 Tf 11.96 0 Td[()E[N]gVar[N] 2(E[N])2=Ologn n!0,whichmeansthatN=(n)w.h.p. Lemma7. Theminimaloodingdelayunderslowmobilitymodelis(p n). Proof. (a)Werstshowthattheminimaldelayis(p n).Fromrandomwalkmodel,nodespeedisupperboundedbyp 2An=m=O(1)andthetransmissionrangerc=(1).Therefore,informationpropagationspeedwillbenolargerthan(1)pertimeslot.ItcanbeshownthatthedistancebetweentheinitialpositionsofS-Dpairis(p An)=(p n)w.h.p.[ 90 ].Hence,theexpecteddelayisatleast(p n)timeslots. 115

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(b)Wethenshowthatthe(p n)delayisachievableusingooding.Wecitethefollowingimportantresultaboutrumorspreadingontorus:Theorem3in[ 82 ]statesthatfollowingtheoodingrulementionedinSection 4.6.1 ,attimeslott,thereexistsasub-torusofsizep tp t,whereforeachcellinthissub-torus,thereexistsatleastonerednode.Therefore,in(p n)timeslots,wecancoverthewholetorusofsizep np nw.h.p. 4.6.2Performanceof2-HopRelaywithRLCInthissubsectionweproveTheorem 4.4 forScheme1.DelayofScheme1underSlowMobilityModel:WeconsideradecoupledversionofScheme1,whichconsistsofthreedecoupledphasesintimeaxis:(1)Thesourcenodesuccessfullytransmitsmcodedpacketsofitscurrent(big)generationtomdistinctrelaynodes.IttakesN1timeslots,andobviouslyN1m=(n).(2)ThesemrelaynodestakeindependentrandomwalkswhichtakesN2timeslots.Afterthisphase,mrelaynodeswillbeuniformlydistributedinthetorus.(3)Thedestinationnodecollectsk=(n)codedpacketsfromthenetwork.IttakesN3timeslotsandobviouslyN3k=(n).ObviouslyD(n)=N1+N2+N3,andinwhatfollowswewillproveD(n)=(n)byshowingthatN1=N2=N3=(n).Notethatinsteadofcollectingcodedpacketsassoonaspossible,herethedestinationnodebegintocollectpacketsafterN1+N2timeslots.ObviouslythisstrategyisnotasefcientasScheme1.However,itsdecouplingnatureleadstoanalyticaltractability.WewillshowthattheinefciencyintroducedbythisdecouplingstrategywillnotchangetheorderofthedelayofScheme1.Phase1:FromLemma 5 ,wedirectlyobtainthatN1=(n). 116

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Phase2:From[ 7 ],weknowthatthemixingtimeofasimplerandomwalkonap np ntorusisalso(n).Therefore,thereexistaconstant"suchthatafterN2="n=(n)timeslots,thesemnodeswithcodedpacketsareuniformlydistributedinthetorusw.h.p.whichmeansthateachnodeinthenetworkhascodedpacketswithaconstantprobability.Phase3:Giventhatmrelaynodesareuniformlydistributedoverthetorus,hereweneedtoprovethatafterN3=(n)timeslots,thesourcenodecancollectkdistinctcodedpackets.RecallthatLemma 3 showsthatthedestinationnodewillbescheduledasthereceiverwithanon-vanishingprobabilitypsineachtimeslot,whichisindependentofn.LetN3`denotethenumberoftimeslotsrequiredbythedestinationnodetomeetmdistinctrelaynodes,wehavethatN3`=(N3).Therefore,weonlyneedtoprovethatN3`=(n).RecalltheproofinLemma 5 ,weknowthatafterN3`=(N3)timeslots,thesourcenodewillmeetk=(n)distinctrelaynodesandobtainkdistinctcodedpacketsw.h.p.ThroughputofScheme1underSlowMobilityModel:Accordingtoabovediscussion,thereare(n)nodesthatcanhavesuccessfultransmissionsimultaneouslyatanytimeslot.Considerthetransmissionofpacketsfromrelaystodestinations.LetA(i,t)bethenumberofcodedpacketsreceivedbythedestinationd(i)intimeslottandA(t)=Pni=1A(i,t)bethetotalnumberofcodedpacketsreceivedintimeslott,wehaveE[A(t)]=(n).Notethatthemobilerandomnetworkisanirreduciblenite-stateMarkovchainandA(t)isaboundednon-negativefunctionofthestateofthisMarkovchainattimet.ThereforebytheergodicityofsuchaMarkovchain,limT!11 tTXt=1A(t)=E[A(t)]=(n).Thusthetotalrateatwhichcodedpacketsaretransmittedfromfromrelaystodestinationsis(n).Fromthesymmetryofthenodesandtherandomnessoftheschemeitfollowsthateachofthendestinationsreceivesatrateof(1)ofcoded 117

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packets.Sinceeachcodedpacketcontainsinformationofm=n=(1)originalpackets,thethroughputisstill(1)inoriginalpackets.DelayofScheme1underFastMobilityModel:WeconsideradecoupledversionofScheme1,whichconsistsoftwodecoupledphasesintimeaxis:(1)Thesourcenodesuccessfullytransmitsmcodedpacketsofitscurrent(big)generationtomdistinctrelaynodes.IttakesN1timeslots,andobviouslyN1m=(n).(2)Thedestinationnodecollectskcodedpacketsfromthenetwork.IttakesN2timeslotsandobviouslyN2k.Phase1:FromLemma 4 ,wedirectlyobtainthatN1=(n).Phase2:Forthedestinationnodetomeetk=(n)distinctrelaynodes,itisastandardcouponcollectorproblemwhichrequiresthatN2=(nlogn).WhenRLCisused,itisprovedin[ 33 43 ]thatthedestinationnodeonlyneedstocollect(n)codedpackets,whichrequiresthatN2=(n).ThereforethetotaldelayD(n)=N1+N2=(n).ThroughputofScheme1underFastMobilityModel:In[ 121 122 ],itisshownthatthecapacityregiondependsonlyonthesteadystatenodelocationdistribution.Notethat,forbothi.i.d.andrandomwalkmobilitymodels,insteadystatenodesareindependentlyanduniformlydistributedoverthetorus.Therefore,agivenschemewillachievethesamethroughputunderthesetwodifferentmobilitymodels.Recalltheresultprovedunderrandomwalkmobilitymodel,wehaveT(n)=(1)underi.i.d.mobilitymodelalso. 4.6.3PerformanceofMulti-HopRelaywithRLCInthissubsection,weproveTheorem 4.5 forScheme2.DelayofScheme2underFastMobilityModel: 118

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WeconsideradecoupledversionofScheme2,whichconsistsoftwodecoupledphasesintimeaxis:(1)Thesourcenodesuccessfullytransmitsm=(logn)distinctcodedpacketsofitscurrent(big)generationtorelaynodes.AccordingtoLemma 3 ,itonlytakesN1=(logn)time-slotsonaverage.Afterthisphase,accordingtotheoodingschemeanalyzedinLemma 4 w.h.p.eachnodewillhaveatleastonecodedpackets.Obviously,somerelaynodesmayhavethesamecodedpackets.However,basedonthepropertyofmobilitypattern(i.i.d.),theinformationcontainedink=(logn)distinctcodedpacketsisuniformlydistributedoverthetorus.(2)AccordingtoProposition 4.2 ,thedestinationnodecollectskcodedpacketsandrecoverstheoriginalkpacketswithin(N2)=(logn)timeslots.ThroughputofScheme2underFastMobilityModel:Notethatinordertoenable(logn)delayinthegeneralcasewherenS-Dpairsareactiveandsharethenetworkresources,weapplyaoodingprotocolinScheme2inwhichtheoldestgenerationthathasnotbeendeliveredtoallnodesisselectedtodominatenetworkresources.Scheme2isfairinthatincaseofties,sessionipacketsaregiventoppriorityeveryntimeslots.Sincewegetk=(logn)originalpacketsin(k)timeslots,thethroughputforthephasewhenthetransmissionsofthisS-Dpairdominatethenetworkis(1).Forfairnessembeddedinthescheme,thissituationhappensoncefor(1=n)phases.Therefore,thelong-termthroughputisT(n)=(1=n).DelayofScheme2underSlowMobilityModel:WeconsideradecoupledversionofScheme2,whichconsistsofthreedecoupledphasesintimeaxis:(1)Thesourcenodesuccessfullytransmitsk=(p n)distinctcodedpacketsofitscurrentgenerationtorelaynodes.AccordingtoLemma 3 ,itonlytakesN1=(p n)time-slotsonaverage. 119

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(2)RelaynodeswithcodedpacketstakeindependentrandomwalksandperformpacketmixingaccordingtoRLCrules.Notethat,unlikei.i.d.mobilitymodel,whichwillchangethenodepositioninaglobal-scaleineachtimeslot,itisnotsoobviousthatwhetherafterN2=(p n)timeslots,theinformationcontainedink=(p n)distinctcodedpacketsisuniformlydistributedoverthetorusunderrandomwalkmobilitymodel.RecallthatithasbeenprovedinLemma 7 thatbasedonouroodingschemeandRLCrules,w.h.p.after(p n)timeslots,everygivenpacketcanspreadoverthewholetorus.Therefore,wecanndaconstant"suchthatafterN2="ntimeslots,theinformationcontainedink=(p n)distinctcodedpacketsisuniformlydistributed.(3)AccordingtoProposition 4.2 ,thedestinationnodecollectskcodedpacketsandrecoverstheoriginalkpacketswithin(N2)=(p n)timeslots.ThroughputofScheme2underSlowMobilityModel:Theproofissimilartotheproofunderfastmobilitymodel.Theonlydifferenceisreplacingk=(logn)intheaboveproofwithk=(p n)forslowmobilitymodel. 4.7ChapterSummaryInthischapter,wecharacterizethethroughput-delay-storagetradeoffsinmobileadhocnetworks(MANETs)withnetworkcoding,andcompareswiththescenarioswhereonlyreplicationandforwardingareallowedineachnode.Theschemes/protocolsachievingthosetradeoffsinaneffectiveanddecentralizedwayareproposed.Thescenariosinwhichnetworkcodingprovidessignicantimprovementonnetworkperformanceareidentiedunderdifferentnodemobilitypatterns.TheinsightsonwhenandhowinformationmixingisbenecialforMANETswithmultipleunicastandmulticastsessionsareprovided. 120

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CHAPTER5PROVIDINGINCENTIVESINMULTI-HOPWIRELESSNETWORKSWITHNETWORKCODING 5.1ChapterOverviewInamulti-hopwirelessnetwork(MWN),whenthesourceandthedestinationnodesforapacketarenotwithindirecttransmissionrangeofeachother,theymustrelyonintermediatenodestoforwardpacketsbetweenthem.Hence,theperformanceofaMWNheavilydependsontheparticipatingnodes'willingnesstocooperate.Ifallnodesarecooperative,suchasinmilitarynetworksconguredtobehavecorrectlybyacentralauthority,thencooperationcanbetakenforgranted.However,formostcurrentandemergingMWNs,participatingnodesareownedandadministeredbydifferentauthoritiessuchasdifferentpersons,andthereforeareautonomous.Whenanodeforwardingtrafcforothernodes,itexpendsitsownbandwidthandpowerresourcewithoutanydirectbenet.Aself-interestednodethereforehasastrongincentivetofreeride,i.e.,usethenetworkresourcesofothernodeswithoutcontributingitsown.Iffree-ridingbehaviorsprevail,suchnetworksevencannotfunction.Therefore,theproperdesignofincentivemechanismforencouragingresourcesharingatthenetworklayerisessentialforthesuccessofanyMWNincivilianorcommercialenvironments[ 20 ].Obviously,theinteractionsamongautonomousandself-interestedentitiescanbemodeledandanalyzedasasocio-economicsystem,andhowtostimulatecooperativebehaviorsinsuchasystemisanextensivelystudiedtopicinsociologyandeconomicswitharichcollectionofanalyzingtechniquesandpromisingsolutions[ 32 117 ].Therefore,itisnotsurprisingthatallproposedincentivemechanismsforMWNsintheliteraturedrawanalogiesfromtheircounterpartsinhumansocieties.Table 5-1 givesaglimpseofdesignspaceofincentivemechanismsforMWNsandpointsouttheirrelationshipswitheconomicandsocialmechanismsenablingcooperationsinhumansocieties.Existingapproachesforprovidingincentivesbasicallycanbeclassiedintothreecategoriesasfollows. 121

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Table5-1. Thedesignspaceofincentivemechanisms EconomicmechanismsMechanismsusedinwirelessnetworksSocialmechanisms BarteringBartering(tit-for-tat)[ 19 51 115 142 167 ]DirectreciprocityIndirectbarteringNetworkedbartering[ 107 ]Generoustit-for-tat[ 147 ]IndirectreciprocityCommoditycurrency? Generoustit-for-11 Reputation 1111 [ 18 75 111 ] Generoustit-for-1 ReputationFiatcurrencyVirtualcurrency[ 31 135 158 180 181 ] Note:Ifthereaderfeelsuncomfortablewiththistable,pleasedonotblametheauthor.Theauthorofthisdissertationalwayswantstoofferthereaderaclearandvividpresentationofhisidea.ButtheGraduateSchooloftheUniversityofFloridahasverystrictanduniformformattingrequirementsfordissertations.Forexample,noverticallinesareallowedforanytableandyoucanonlyhavethreehorizontallinesforonetable.TheauthorhasalreadytriedhisbesttocommunicatewiththeGraduateSchoolEditorialOfce,butobviouslyasaninternationalstudenthecannotghtthesystem.ThegoodnewsisthatthischapterhasalreadybeenpublishedbyIEEE[ 172 ],sopleasereferto[ 172 ]fortheoriginaltable.Ifthereaderthinksthatthistableisinfactmuchbetterthantheoriginalone,thenpleasethankthegreatworkofourGraduateSchoolEditorialOfce. Therstcategoryisbarterbasedapproaches,whicharebasedondirectreciprocity:nodeAwouldprovideresources/servicesfornodeBonlyifBsimultaneouslyprovidesresources/servicesfornodeA.Thiskindofbilateralandsynchronousresource/servicetradingmakesbarterextremelysimpletoimplement.Fromasystemperspective,thereisnoneedtokeepanylong-termstateinformation,intheformofeitherreputationorcurrency,andasaconsequencetheimplementationcostofbarterisalmostzero.However,synchronoustradingiseasytofailwhenanactionanditsrewardarenotsimultaneous.ThisistrueforMWNs:theactionispacketforwardingandthebenetisbeingabletosenditsownpacket.Therefore,bartersingeneralcannotprovidesufcientincentivetosustainfullcooperationinaMWN.Thesecondcategoryisvirtual-currencybased,inwhichparticipatingnodeswouldearnvirtualcurrencybyprovidingresources/servicestoothersandspendthevirtualcurrencytoobtain 122

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resources/servicesfromothers.Bytakingvirtualcurrencyasamediumofexchange,nodescanthentraderesources/servicesasynchronously,whichovercomestheshortcomingofbarters.Virtualcurrency,however,incursahighimplementationoverhead,e.g.,billingande-cashtransfers,implementationsofcentralizedbankandelaboratingdispute-resolutionmechanisms,etc.Inthethirdcategory,i.e.,reputationbasedapproaches,participantsbuilduptheirreputationscoresbyprovidingservicesforothers,andhighlyreputedparticipantsreceivepreferentialtreatmentwhentheyneedhelp.Obviously,reputationscoresherecanbetreatedasanotherformofvirtualcurrency.Therefore,reputationbasedapproachessharethesameadvantagesanddisadvantagesasvirtual-currencybasedones.TheprosandconsoftheseapproachesarealsodiscussedindetailinSection 5.2.1 .InthespectrumofincentivemechanismsasillustratedinTable 5-1 ,rangingfrombarteratoneextremetovirtualcurrency/reputationattheother,thesweetspotforincentivemechanismdesigninMWNslikelyliessomewherein-between.Byexploringthewholedesignspace,weproposeanewparadigm,ControlledCodedpacketsasvirtualCommodityCurrency(C4),forprovidingincentivesinagenericMWN.InourC4,throughintroducingseveraltechniquesfromnetworkcoding,codedinformationpacketsareutilizedasanewkindofvirtualcurrencytofacilitateresource/serviceexchangesamongself-interestednodesinaMWN.ByintroducingC4,wemakethefollowingcontributionsinthischapter:SincethevirtualcurrencyimplementedinC4alsocarriesusefuldatainformation,itisthecounterpartoftheso-calledcommoditycurrencyusedinthephysicalworld.Therefore,ourC4llsthegapinthedesignspace(i.e.,thequestionmarkinTable 5-1 ),andrepresentsanoveldesignparadigm.Byintroducinganewkindofvirtualcurrency,ourC4cansolvefundamentalproblemsstumblingbarters:asynchronicityovertimeandasychronicityovernodes,andatthesametimeonlyincuralowimplementationoverheadcomparabletobarterbased 123

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approaches.Therefore,ourC4hassignicantadvantagesincomparisonwithexistingsolutions.WedevelopmathematicalmodelstostudytheeffectivenessandefciencyofC4.WetheoreticallyshowthatC4isperfectlyefcienttosupportMWNswithbroadcastandmulticasttrafcs.Forpureunicastcommunications,byadjustingcodingparameters,C4providesasystematicwaytosmoothlytradeincentiveeffectivenessforsmallimplementationcost,andtraditionalbarterbasedandvirtual-currencybasedschemesarejusttwoextremecasesofC4.Whenthesocialcontactinformationamongmobileusersisavailable,weproposetwotechniquestofurtherreducetheimplementationcostsofC4withoutsacricingincentiveeffectivenessforpureunicastcommunicationsinMWNs.Therestofthischapterisorganizedasfollows.RelatedworkissummarizedinSection 5.2 .ThedesignofC4foragenericMWNisdescribedinSection 5.3 .InSection 5.4 weevaluatetheperformanceofC4throughtheoreticalanalysisandsimulations.InSection 5.5 ,weshowhowtoutilizethesocialnetworkformedbymobilenodestofurtherimprovetheperformanceofC4withpureunicastcommunications.ConclusionandfutureworkaredescribedinSection 5.6 5.2RelatedWorkandMotivationforC4Inthissection,werstpresentasystematicandhistoricaloverviewofincentivemechanismsforMWNs.ThenthedesignofourC4ismotivated.Moreover,theincentiveproblemsofnetworkcodingitselfisalsoinvestigatedinthissection. 5.2.1ExistingIncentiveMechanismsforMWNsAsmentionedinSection 5.1 ,wecanuseasocio-economicsystem(i.e.,amarket)tomodelresource/serviceexchangesamongindividualnodes,andanyinteractioninthissystemcanbedecomposedintoasetofelementaryinteractions(i.e.,tradesillustratedinFigure 5-1 )betweentwonodes.Thetimespentinsearchingfora 124

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Figure5-1. Trademodelsforelementaryinteractionsbetweentwonodes.Notethatthearrowhereonlyrepresentstheserviceprovider-receiverrelationshipbetweentwonodes,notthedatapackettransmission.Whentheprovidedserviceisforwardingdatapackets,thenthereceiverofthisserviceisthetransmitterofdatapackets.Whentheprovidedserviceiscontentdistribution,thentheproviderofthisserviceisthetransmitterofdatapackets. successfultradeisdenedastransactioncostineconomics[ 117 ],whichshouldbeminimizedbyaneffectiveincentivemechanism.[BarterBasedSchemes]Barter(alsocalleddirectreciprocityinsociology),abilateralandsynchronousexchangeofservices/resources(cf.Figure 5-1 (a)),istheoldestandsimplesttradingforminhumanhistory[ 117 ].Abartercanbesuccessfulonlyifinvolvedtwopartiessatisfyadoublecoincidenceofwants,i.e.,theyaremutuallyinterestedineachother'sservices/resources.Purebarter-likeschemeisrstproposedforcontentdistributionapplicationsinwirelessnetworks[ 51 167 ],e.g.,sharinglesofapopularmovieamongmobilenodes.Becauseallnodeshavethecommoninterestinalllesrelatedtothemovieandeverynodeonlyhaspartofles,whentwonodes 125

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meettheycanexchangethelestheydonothavewithhighprobability.1However,forotherkindsoftrafcpatternlikemultipleunicasts,barteriseasytofailwhenfacingthefollowingtwoasynchronicities[ 19 107 ]:(i)asynchronicityacrosstime:Twonodesmightnotsimultaneouslyrequireeachother'sforwardingservices.Toovercomethisshortcoming,asynchronousbarter(alsocalledtit-for-tat)isintroducedin[ 115 142 ].AsillustratedinFigure 5-1 (b),eachnoderecordstheinteractionhistorywithothernodesandonlycooperateswiththenodeswhoalsocooperateinthehistory.Asaconsequence,eachnodehastostoreinformationproportionaltothenumberofnodesitinteractedwith,whichisalargeoverheadinalargesystem.(ii)asynchronicityacrossspace:Bartersystemisnottransitive,forthesituationshowninFigure 5-1 (c),ifAlice(A)doesafavortoBob(B),andBobtoCarol(C),AlicecannotcallforafavorfromCarol.Suchasituationcanariseduetotopologicalasymmetryinthepositionsoftwonodes,andisnormalinMWNs.Tofacilitatemultilateraltradinginthissituation,indirect-barterbasedschemeslikenetworkedbartering[ 107 ]andgeneroustit-for-tat[ 147 ]areproposed.Thecommonproblemofthiskindofschemesisthatthecycleinvolvedinindirectbartermaybeverylong,andalargenumberofcontrolpacketsneedtobeexchangedinordertocoordinateallparticipants'behaviors.Infact,currencyorreputationcanbeutilizedasamoreefcientwaytocoordinateparticipants'behaviorsinthissituationanddecomposethiscircleintoelementaryinteractions.Therefore,ifwewanttokeepthelowimplementationcostofbarter-likeschemes,mobilenodesmayneedtospendmoretimetondanothernodewithadoublecoincidenceofwants,whichimpliesahightransactioncostandalowincentiveeffectiveness. 1Althoughstrictsynchronicityisimpossibleforwirelessenvironmentswiththesharedchannel,thistradeisstillsynchronousinthesensethatitistakingplacewithinveryshorttimeperiods,e.g.,ontheorderofseconds. 126

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[Virtual-currencyBasedSchemes]Thehistoryofeconomics[ 32 117 ]hasshownthatcurrencyasamediumofexchangecanfacilitateasynchronousandmultilateraltrading,overcometheshortcomingsofbarters,andthereforeminimizetransactioncost.Virtualcurrencyproposedintheliterature(cf.[ 31 135 158 180 181 ])mimicsitscounterpartinthephysicalworld:participatingnodespossessacertainamountofvirtualcurrency(orcredit),andifanodewantstosendapacket,ithastopayallthenodesparticipatinginrelay.Anuncooperativenodewilleventuallyrunoutofcurrencyandstoptransmitting.Thepaymentcanbedirectbyusingsomeformofdigitalcash(e-cash)orviaacentralauthoritythatservesasabank,asshowninFigure 5-1 (d).Itcanbeeasilyobservedthatforschemeswithe-cash,thenumberofcontrolpacketsfore-cashtransferareonthesameorderofthatofdatapackets.Whencentralizedbankisinvolved,evenmorecontrolpacketsareneededforbilling,disputeresolution,etc.Thisisahugeimplementationoverheadforeveryvirtual-currencybasedschemes.Anothercommonproblemforthoseschemesisthat,sincethevirtualcurrencyiscreatedexnihiloonamarket,itisextremelydifcultifnotimpossibletomaintainanreasonableaveragecurrencylevelwithinthesystem[ 117 ].However,thisiscrucialforincentivestoworkproperly.Iftheaveragecurrencylevelgrowstoohigh,everyonewillberichandnolongerhaveanincentivetocooperate(justliketheinationinrealworld),andconversely,ifthereisnotenoughcurrencywithinthesystemthenhardlyanyonewillbeabletotransmit.[ReputationbasedSchemes]Reputationworksinasimilarwaytocurrency.Weearnourreputationscoresbyexhibitingcooperativebehaviorsandothersdecidewhethertocooperatewithbasedonourreputationscores.InMWNs,asproposedin[ 18 75 111 ],nodesmonitortheirneighboringnodes'behaviors,andgiverst-handreputationscorestoeachother.Ingeneral,rst-handreputationinformationisnotenoughformakingafairjudgementonaparticularnode(notethatwhenonlyrst-handreputationinformationisavailable,thenweobtainasynchronousbarter).Therefore, 127

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second-handreputationinformationneedbepropagatedandintegratedinthesystemlevel.Thereputationscorescanbemanagedlocallyorglobally,asillustratedinFigure 5-1 (e).Forlocalizedapproach,eachnodewillaskitsneighborstogiveareportaboutitsownbehaviors,andthisreportwillbeprovidedwhenthenodeisrequiredtogiveevidencetoproveitsreputation.Forglobalapproaches,atrustedauthoritywillcollectsecond-handreputationinformation,andbroadcasttheintegratedreparationscorestothewholenetwork.Likevirtual-currencybasedschemes,themaintainingandupdatingofreputationinformationinthesystemlevelincurhighimplementationoverhead.Reputationbasedschemesalsohaveseveralinheritedproblemsasfollows.Firstly,reputation-basedsolutionsusuallycannotresistSybilattack,whitewashingattack,andcollusion.Alsoitisdifculttosecuresecond-handreputationpropagation.Secondly,itisalmostimpossibletoevaluatetheincentivesprovidedbyreputationinaformalmanner.Thethirdweaknessinreputation-basedschemesistheirheavyrelianceuponthepremisethatmisbehaviorcanbedetectedbyneighboringnodes.However,toimplementeffectivemisbehaviordetectionschemesamongmobileusersisreallydifcultinpractice. 5.2.2MotivationforOurC4Fromabovediscussions,wecanseethatexistingsolutionsareeitherlesseffectiveorincurhighimplementationcosts,andthereforedonottwellwiththeuniquerequirementsofMWNs.Anewdesignparadigmisneeded.Therstquestionweshouldaskis:Isthereanyroomforfurtherexploration?Ifwegothroughthedesignspaceofincentivemechanisms,theredoexistanunexploredareawhichisthecounterpartofcommoditycurrencyinthephysicalworld(i.e.,thequestionmarkinTable 5-1 ).Economistsdenecurrency(ormoney)asagenerallyacceptablemediumofexchange[ 32 117 ].Withtheuseofcurrency,theproblemofdoublecoincidenceofwantsisavoided,andthetransactioncostofsearchingasuccessfultradeisreduced.Historically,currencyoriginatedascommoditycurrency. 128

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Table5-2. Costanalysisofdifferentincentivemechanisms XXXXXXXXXXXXCostSchemeInthephysicalspaceInthedigitalspaceBarterCCFC/RptBarterC4VC/Rpt TransactionHighLowLowHighLowLowTransportationZeroHighLowZeroLowLowImplementationZeroZeroLowZeroLowHigh Whenaphysicalcommodity(e.g.,compressedtealeavesinancientChina)hasvaluetoeveryoneandisgenerallyacceptedasthepaymentforothergoods/services,itcanserveasacurrency.Thiskindofcurrencyhasintrinsicvalue(i.e.,valuableinitsownright),andstillcanbeconsumedasaphysicalcommodity(i.e.,tomaketea)whennotneededfortrade.Theproblemofcommoditycurrencyliketealeafbricksisthatitisheavyandhardtotransportfromoneplacetoanother.Fiatcurrency(papercurrency)isinventedtoreducethecostsinvolvedinstoringandcarryingcommoditycurrencyintrading(i.e.,transportationcost).Fiatmoneyiswithoutintrinsicvalueasaphysicalcommodity,andderivesitsvaluebybeingdeclaredbyagovernmenttobelegaltender.Obviously,virtualcurrencydiscussedinSection 5.2.1 isexactlythecounterpartofatcurrencyinthephysicalword,becausevirtualcurrencyhasnointrinsicvalue,i.e.,itcarriesnousefuldataandisjustusedasamediumofexchange.So,oursecondquestioncomes:Whatisthecounterpartofcommoditycurrencyinthedigitalworld?Thereareonlytwoconditionsforalikelycandidate:(i)itshouldcarrydatainformationbecausethecommodityinaMWNcanonlybedatapacket;(ii)ithasvaluetoeveryone.Followingtheseclues,thekeyideaofourC4naturallyemerges:codedpacketscanserveasvirtualcommoditycurrencyforMWNs.Byutilizingnetworkcoding,originaldatapacketsfrom/todifferentmobileusersaremixedtoproducecodedpackets.Asaconsequence,eachcodedpackethasvaluetoeveryone,andisreadytoactasvirtualcurrencytofacilitatecooperations.Theonlyquestionleftthenis:Whatarethebenetsofusingvirtualcommoditycurrency(i.e.,C4)inMWNs?Tofairlyevaluateanincentivemechanism,weneed 129

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considerthreekindsofcosts.Inourdailylife,comparedtotheexchangeofphysicalcommodities,theimplementationcostofatcurrency(FC)orthemouth-to-mouthreputation(Rep)canbeignored.AsdescribedinTable 5-2 ,inthephysicalspace:barterhasahightransactioncost;commoditycurrency(CC)hasahightransportationcost;onlyFCcankeepallthreecostslistedinTable 5-2 low,andthereforeisthebestchoice.Thisexplainsthefollowingfact:nearlyallcontemporaryeconomicsystemsarebasedonatcurrency.SowhathappensinthedigitalspacelikeaWMN?Thekeypointhereisthatthegoods/servicesinaWMNareinformationpackets/packettransmissions.Virtualcurrency(VC)andreputationarealsostoredandtransactedintheformofinformationbits.Therefore,comparedtothedatapacketcommunication,theimplementationcostofVC/Repcannotbeignoredandinfactisprettyhigh.TheimplementationcostofourC4isalwayssmallerthanthatofVC,becausewhene-cashisusedasthemediumofexchange,itonlyrepresentscontroloverhead;whilewhencodedpacketisused,italsocarriesusefuldata.Unlikethephysicalcommodity,transportationcostsofcodedpacketsaresmall.Therefore,fromTable 5-2 ,wecanseethatC4isthebestchoiceinthedigitalspace. 5.2.3NetworkCodingandIncentivesThebenetsofutilizingnetworkcodinginwirelessnetworks,suchasimprovingthroughput,reducingenergyconsumption,simplifyingtransmissionscheduling,etc.,havebeenextensivelystudiedintheliterature(cf.[ 170 ]andreferencestherein).However,asfarasweknow,ourC4isthersttodemonstrateanewpotentialbenetbroughtbynetworkcodingforwirelessnetworks,i.e.,providingincentivesforcooperation.Incentiveproblemsinwirelessnetworkswithnetworkcodinghavealsobeenstudiedin[ 28 176 ].Bothofthemapplygametheorytostudycodingstrategiesadoptedbyindividualnodesandstillusevirtualcurrency(credit)tocreateincentives.InourC4,weassumethatthemainapplicationofMWNsistoprovideInternetaccessservices,andmosttrafcinaMWNwillgothroughtheinfostation(denedbelow),which 130

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enforcesrandomlinearnetworkcoding(cf.Section 7.3.1 )toimprovethewholesystem'sefciency.Therefore,noroomisleftforself-interestedmobilenodestochangecodingstrategies,andcodedpacketscanbesafelyutilizedtocreateincentives.Bytreatingnetworkcodingasatooltoproducevirtualcommoditycurrency,ourC4providesaneffectiveandlightweightsolutiontoinducecooperation,whichisimpossiblefor[ 28 176 ]basedontraditionalvirtualcurrency. 5.3DesignandImplementationofOurC4Inthissection,wedescribethedesignofourC4.Werstclarifythetargetscenariosandbasicassumptions. 5.3.1SystemModelandProblemFormulationBasedonthecommonrequirementsoffuturemobilecommunicationenvironment[ 5 72 103 162 ],ourC4assumesthefollowinggenericmodelforMWNs.AsillustratedinFigure 6-1 (a),therearetwokindsofentities:mobilenodesandinfostations.Mobilenodesarecontrolledbyautonomousandself-interestedclientsandareinterestedinInternetaccessservices.Amobilenodecanestablishashort-rangewirelesslink(e.g.,Wi-Fi)withothermobilenodesinitsvicinity.Theshortrangelinkstendtobeintermittentbecauseofnodemobility.InfostationsaremanagedbythesystemoperatoranddirectlyconnectedtotheInternetwithreliableandhigh-bandwidthlinks(i.e.,backbonelinks).Aninfostationcanusealong-rangelow-bandwidthradio(e.g.,cellularinterface)toconnectwitharemotemobilenode(butwedonotassumethatallmobilenodesarecoveredbyinfostations),oruseashort-rangewirelesslinkwithhighdataratetoconnectwithaclosemobilenode.Infostationsarethedatasourceswithinthewirelessdomainandwanttoprovidebetterservicestoclients.AsillustratedinTable 5-3 ,thisgenericmodelincludesseveralimportantmulti-hopwirelessnetworkarchitectureswhichattractgreatinterestfrombothacademiaandindustry. 131

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Figure5-2. Agenericarchitectureformulti-hopwirelessnetworks. Weassumethatallmobilenodesareself-interestedandstillrational.Theyhavethenon-cooperativebehaviorsmainlybecausetheywanttosaveresourcesuchasbandwidthandbatterypower.Wealsoassumethatallinfostationsareunderthecontrolofoneauthority,andtheywilldoalltheycantoencouragemobilenodestouseshort-rangelinks.Unlikepreviouswork[ 18 28 31 75 111 135 158 176 180 181 ],ourC4doesnotmakeanyassumptionsaboutroutingprotocolsusedinMWNs.OurC4isdesignedtosupporttraditionalstore-and-forwardroutingschemesaswellasnewroutingschemesinaDTNfashion(i.e.,store-carry-and-forward[ 178 ]).AlsoourC4allowsallpossiblecombinationsoftrafcpatterns(e.g.,broadcasts,multicastsandunicasts).Theonlyrequirementisthatmosttrafcsgothroughinfostations. 5.3.2MethodologyofOurC4WetakeonebroadcastsessionfromtheinfostationtomobilenodesasanexampletoillustratethebasicdesignofourC4. 132

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Table5-3. Examplesofmulti-hopwirelessnetworks Multi-hopwirelessnetworkInfostationMobilenode Multi-hopcellularnetwork[ 103 ]BasestationMobilestationWirelessmeshnetwork[ 5 ]MeshrouterWirelessclientMobilesocialnetwork[ 72 ]ServiceproviderMobileuserVehicularadhocnetwork[ 162 ]Roadsideinfo.unitVehicle SupposethisbroadcastsessionisaboutdistributingamoviestoredinawebserverintheInternet.Theinfostationtreatsallnetworklayerpacketsreceivedfromthewebserverasoriginaldatapackets(OPs).WeassumeeachOPhaslbytes.Attheinfostation,randomlinearnetworkcoding(RLNC)[ 67 ]isappliedtoanitesetofkOPs(i.e.,OP1,OP2,...,OPk),whichiscalledageneration.TheneachgenerationcanberegardedasaklmatrixOP,withrowsbeingthekOPsofthegeneration,andcolumnsthelbytesofeachOP.TheencodingoperationproducesalinearcombinationoftheOPsbyCB=CVOP,whereCVisankkmatrixcomposedofrandomlyselectedcodingcoefcientsintheGaloiseldGFqofsizeq.Thecodeddatablocks(rowsinCB)andthecodingvectors(rowsinCV)areconcatenatedasthecodeddatapackets(CPs).Forexample,thecodeddatablockCBj=Pki=1CVjiOPiwhereCVji2GFqandthecodedpacketCPj=CVjkCBj.TwocodedpacketsCPiandCPjarecalledindependentifCViandCVjareindependentvectors.InourC4,insteadofsendingOPs,theinfostationsendsCPstomobilenodeswithshort-rangelinks.Thedecodingoperationatthedestination(i.e.,mobilenodes),initssimplestform,isthematrixmultiplicationOP=CV)]TJ /F9 7.97 Tf 6.59 0 Td[(1CB,whereeachrowofCBrepresentsacodeddatablockandeachrowofCVrepresentsthecodingvectorsaccomplishedwithit.ThesuccessfulrecoveryoftheoriginalpacketsOPrequiresthatthematrixCVbeoffullrank,i.e.,thedestinationmustcollectkindependentCPs.Eachmobilenodecaneitherdownloadpacketsfromaninfostationorexchangepacketswithneighboringmobilenodes.Inanon-cooperativenetworkwithoutanyincentivemechanisms,theformeristheonlymechanismforpacketdissemination.Itonlyusesthehigh-speedchannelbetweenaninfostationandanodenearit,while 133

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Figure5-3. Acomparisonbetweenbartering(withoutcoding)andourC4. wastingalltheequallyexcellentchannelsbetweenneighboringmobilenodes.Abarter-likescheme(withoutnetworkcoding)alleviatesthisproblemasfollows:whentwomobilenodescontact,theyinspectthepacketcontentsofeachother.Ifeachnodeidentiesapacketthatitwants,abilateralpacketexchangetakesplace.However,evenforbroadcasts,nodescaneasilyendupinadeadlocksituationbecauseoftherequirementofmutualwants.InourC4,letSu(t)andSv(t)denotethesubspacesspannedbytheCPsatneighboringnodeuandv,respectively,atthebeginningofthetimeslott.IfSu(t)*Sv(t),wesaynodevwantsCPsatu.Ifthereexistsmutualwants,thennodeuandvcansuccessfullyexchangeCPsasfollows:nodeusendsanewCPpu2Su(t)andpu=2Sv(t),andnodevsendsbackanewCPpv2Sv(t)andpv=2Su(t)asreturn.ThedesignofC4reliesontwouniquefeaturesofRLNC:RLNCcangreatlyimprovethepossibilityofsuccessfulexchange.AsprovedinLemma2.1in[ 33 ],whenRLNCisused,theprobabilitythattworandomlyselectednodeshavethemutualwantstendsto1whenq,thesizeofGFq,islargeenough(e.g.,q=28).Figure 5-3 illustratesthiseffect.Supposethereareonlytwomobilenodes,andthreeOPsneedbebroadcastedattheinfostation.ItiseasyfortwonodesrunintoadeadlocksituationasexempliedinFigure 5-3 (a).Whenvirtualcurrencyisintroduced,nodeBcanbuypacket2fromnodeAbysendingbackane-cashattimeslot3.However 134

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onlyonedatapacketistransmitted,ande-cashtransferisoverhead.BymixingtheOPs,becausewithhighprobabilityeachCPbringssomenewinformation,itcanbetreatedasvirtualcommoditycurrency,i.e.,generallyacceptableinpayment.Forexample,inFigure 5-3 (b),attimeslot3,nodeBcanbuyapacketfromAbypayingaCP(0,1,1)X.Aftertimeslot3,nodeAcandecodeallOPs,andnodeBonlyneedonemoreCP.Highexchangeefciencyachieveswithoutincurringanyoverhead.RLNCcanserveasafreecipher.BeforethedestinationobtainsenoughindependentCPs,itcannotrecoverallOPs,eventhougheveryCPcontainssomeinformationabouteveryOP.Therefore,RLNCcanserveasafreecipherforatimeperiod.Thisfunctionalityisextremelyusefulforsupportingunicasts.AlsotakeFigure 5-3 (b)asanexample.NowsupposethreeOPsaredestinedfordifferentmobilenodes(AorB).BeforenodeAreceivesthreeindependentCPs,hecannottellwhichpacketisdestinedtowhom.Therefore,evenwhenthreeOPsarealldestinedtonodeB,nodeAstillhasincentivetoparticipateinexchangesbeforehecollectsthreeindependentCPs.Inthisway,weenforcenodeAtohelprelayoneCPtonodeBattimeslot3,whichisimpossibleforbartering. 5.3.3ImplementationDetailsofOurC4Accordingtosource-destinationrelationshipwithinthewirelessdomain,alloriginaldatapacketsinourC4canbeclassiedintotwocategories:(1)downloaddatapackets:thepacketsfromtheinfostationtomobilenodes;and(2)uploaddatapackets:thepacketsfrommobilenodestotheinfostation.OurC4takethewholenetworklayerpacketasanoriginaldatapacket(includingpacketheader).EachoriginaldatapackethasauniquepacketID,andasessionIDindicatingwhichsessionthispacketbelongsto.Amobilenodecantellwhetherhe/sheisinterestedinitbycheckingitssessionID.InourC4,everytransmittedpacketwithinthewirelessdomainisacodedpacket,andhasaspecialformat.Attheinfostation,somedownloaddatapacketsareselectedandcodedtoproducevirtualcommodity 135

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currency(VCC).WecallthesecodeddatapacketsasVCCpackets.AsillustratedinFigure 5-4 (a),acodedpacketconsistsofaheaderandabody.Thebodystoresthecodeddatablock.ForaVCCpacket,theheaderconsistsofthreeparts:(1)TPEeld,whichissetto11toindicateaVCCpacket;(2)sessioneld,whichindicatessessionIDsinvolvedinthecodeddatablockinthebodypart,anditcanbeimplementedbyaBloomlter;(3)coding-vectoreld,wherekisthegenerationsize,pisthenumberofOPsencodedinthepacketandc iisthecodingcoefcientrelatedtotheOPwithpacketIDoid ifor1ip.UnliketraditionalRLNCpacket,herek6=p.Thereasonwillbeexplainedalittlelater.AVCCpacketCPVCCisvalidforamobilenodeu,onlyifuisinterestedinatleastonesessioninvolvedinCPVCCanddoesnocollectenoughcodedpacketstorecoverallOPsofthissession.Download/uploaddatapacketswillnotbedirectlytransmittedinourC4.TheywillbecarriedbyVCCpackets.ForadownloaddatapacketwithpacketIDoid qanddestinationnodeIDd id,whichisnotselectedtoproduceVCCpacket,theinfostationrstselectsoneCPVCCasthecarrier,andthencombinesthesetwopacketsasshowninFigure 5-4 (b).TheTPEeldissetto01toindicateacodeddownloadpacket.ThecodedblockisthelinearcombinationofthecodedblockofCPVCCwiththecodingcoefcient1andthedownloaddatapacketwiththecodingcoefcientc q.AtamobilenodewithnodeIDs id,thecodeduploadpacketisconstructedinasimilarway,asillustratedinFigure 5-4 (c).TheTPEeldissetto00toindicateacodeduploadpacketandthesourcenodeIDs idisalsoincludedinthepacketheader.ThereareonlytwokindsofelementaryinteractionswhentwomobilenodesuandvcontactinourC4:(i)exchangesofcodedpackets:Whennodeuhasacodedpacketwhichvisinterestedin,andthesamethinghappenedfornodev,thenuandvhaveincentivestoexchangethesepackets.(ii)exchangesofrelayserviceandVCCpackets:Whennodeuwantsnodevtorelayanupload/downloaddatapacket,andnodeuhasatleasttwovalidVCCpacketsforv,theninthisinteraction,nodeusendstwopacketsto 136

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Figure5-4. PacketformatinourC4. v:thecodedupload/downloadpacketneedstoberelayedandonevalidVCCpacketasthepayment.Theupload/downloaddatapackethereshouldbecarriedbyanothervalidVCCpacketforv.Notethatiftheupload/downloaddatapacketisdirectlysenttov,thennodevcandropthispacketalittlelaterwithoutbeingdetected.BycombiningthedatapacketwiththeVCCpacket,ourC4providesincentivesforrelaynodevtokeepthisdatapacketbeforevcandecoupleitfromtheVCCpacket.Bydecomposingallpossibleend-to-endcommunicationsintoaseriesofelementaryinteractions,ourC4putsnoconstraintsontheunderlyingroutingprotocolsortrafcpatterns,andthereforeachievesthemaximumexibilityinsupportingdifferentapplicationscenariosinMWNs.ThecodingstrategyadoptedbyinfostationstogenerateVCCpacketsdeterminestheperformanceofourC4.ThekeypointishowtoselectsessionsthatneedbemixedingeneratingVCCpackets.Thereexistsatradeoff:ifwemixthesessionsinterestedbydifferentmobilenodes,wecanobtainVCCpacketswhicharevalidforalargerpopulationofmobilenodes.ThentheVCCpacketscanachieveabetterexchangeefciency.However,theusefuldatainformationcontainedintheVCCpacketforamobileuserisalsodecreased,whichmeanstheoverheadofusingVCCpacketsbecomeslarger.Therefore,acautiouschoiceshouldbemadeinordertoobtainanoptimizedbalance.AquantitativeanalysisofoptimizedcodingstrategieswillbepresentedinSection 5.4 indetail.Here,wejustgivetwoqualitativeguidelines.(i) 137

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Becauseallusersareinterestedinbroadcastdatapackets,theycanbesafelymixedwithoutworryingaboutoverhead.TheonlyproblemisthatinordertoprolongthevalidtimeofVCCpackets,thesizeofonegeneration,i.e.k,shouldbelargeenough.FortraditionalRLNC,thismeansthesizeofthepacketheaderisprettylarge,becausethesizeofcodingvectorwhichshouldbeincludedinthecodedpacketheaderis(k).Toavoidthisproblem,ourC4utilizessparseRLNCproposedin[ 125 ]tocontroloverhead.Givenkoriginaldatapacketsasonegeneration,sparseRLNCrstrandomlychoosespdatapackets,thenperformsRLNContheseppacketstogenerateacodedpacket.ItcanbeshownthatthesuccessfulexchangeprobabilityisonthesameorderofthatofRLNC,whilethesizeofthepacketheaderis(p)wherepk.NotethatourpacketformatshowninFigure 5-4 alreadyincludestheuniquefeatureofsparseRLNCbyembeddingk,pandOPIDsintocoding-vectoreld.(ii)WhenonlyunicasttrafcsexistinaMWN,wehavetomixOPsfordifferentuserstogeneratevalidVCCpackets.WhenkOPsfordifferentsessionsaregroupedforonegenerationandcodedwithsparseRLNC,thesessioneldinaVCCpacketisdeterminedbykOPsinthegeneration,notbypOPsselectedforthisVCCpacket.However,itispossiblethataVCCpacketdoesnotcontainanyusefuldatainformationforamobileuser,butthatuserstilltreatsitasvalid.Amobilenodeevencannotndthisbeforehe/shecanrecoveralloriginalpackets.Therefore,theinfostationcanintentionallydothistoprovideincentivesoroptimizethewholesystem'sperformance.OurC4onlyrequiresthelong-termfairnessamongmobilenodesandutilizesthefreecipherfunctionalityprovidedbyRLNCtoenforcethiskindofoperations. 5.4PerformanceAnalysisofOurC4 5.4.1NetworkModelforPerformanceAnalysisInordertomakethequantitativestudypossible,wedetailournetworkmodelproposedinSection 7.3.1 asfollows. 138

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[CellModel]ConsiderasquaregeographyofareaAnwithaxedinfostationatthecenter,asshowninFigure 6-1 (b).Weassumethegeographywrapsaroundeachboundary,effectivelycreatingatorus.Werefertothistorusasacell.Acellisintendedtomimicatypicalmulti-infostationnetworkinwhichaninnitegridofinfostationspopulateaninniteplane.TheareaAnrelativetothesingleinfostationservestocharacterizethedensityofxedinfostationsovertheterrain.[MobilityModel]Thecellispopulatedwithnmobilenodeswithindependentmobilityprocessesasfollows.Werstdividethecellintom=(n)squareletsofareas2neach,resultinginatwo-dimensionalp mp mdiscretetorus(weassumep misaninteger).Weassumethetimeisslottedandeachnodeindependentlyperformsasimplerandomwalkonthetwo-dimensionalp mp mdiscretetorus,i.e.,letanodebeinsquareletsattimeslott,then,attimeslott+1,thenodeisequallylikelytobeinthesamesquareletsoranyofthefouradjacentsquarelets(i.e.,up,down,left,andrightsquarelets).[ProtocolModel]Forcharacterizingtheconditionforasuccessfulwirelesstransmission,weadopttheprotocolmodelproposedin[ 60 ].Weassumethatallmobilenodes(includingtheinfostation)useacommonrangernfortheirshort-rangetransmissions,andatransmissionfromnodeitonodejissuccessfulifandonlyifdijrnanddkj(1+)rnforanyothersimultaneoustransmitter,saynodek.Here,dijisthedistancebetweennodesiandj,andisapositiveconstantindependentofn.Weassumethatwhentwonodesareinthesamesquarelet,theyareneighbors,i.e.,theycanestablishashort-rangewirelesslink.Therefore,rn=p 2sn.[TransmissionScheduling]Asquareletiscalledactiveattimeslott,iftheMACschedulingschemeallowsonenodeinthatsquarelettotransmitattimeslott.Basedonprotocolmodel,wecanguaranteethatthereexistsaninterference-freeschedulesuchthateachsquareletbecomesactiveregularlyonceinLtimeslotsanditdoesnotinterferewithanyothersimultaneousactivesquarelets.HereLdependsonlyon,and 139

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isindependentofn(cf.Proposition1inourpreviouswork[ 171 ]).TheshadedsquareletsinFigure 6-1 (b)illustrateanexampleofagroupofsimultaneousactivesquareletswhenL=9.Foranactivesquarelet,atmostonenodepairwillbescheduledtocommunicate.Whenthesquareletwiththeinfostation(i.e.,centralsquarelet)isactive,theinfostationwillbeonepartyofthatnodepairandanotherpartywillberandomlychosenfromallnodesinthatsquarelet.Foranyothersquarelet(i.e.,regularsquarelet),whenitisactive,anodepairisrandomlyselectedfromallpossiblenodepairsinthatsquarelet.ThisisintendedtomimicthebehaviorsofIEEE802.11-likeMACprotocol.Foreachtimeslot,weassumeonlytwopacketscanbetransmitted,i.e.,justenoughforonesuccessfulelementaryinteraction.WenotethatabovemodelisthestandardnetworkmodelwidelyusedintheliteratureforMWNperformanceanalysis(referto[ 58 60 171 ]andreferencestherein),anditsbehaviorsarecharacterizedbythefollowingLemmawhenn!1. Lemma8. (1)Givenrn=p 2An=m=(p An=n),thenetworkformedbyshort-rangewirelesslinksisdisconnected,i.e.,theredoesnotexistacontemporaneouspathbetweentworandomselectednodeswithhighprobability.(2)Foragiventimeslot,theprobabilitythatamobilenodeisintheactivecentralsquareletandselectedintothecommunicationpairispI= mL=(1=n),whereisaconstant.(3)Foragiventimeslot,theprobabilitythatamobilenodeisinanactiveregularsquareletandselectedintothecommunicationpairis,andisaconstant,i.e.,independentofn. 5.4.2PerformanceAnalysisforBroadcastandMulticastTrafcsWerstconsiderbroadcastscenarios.SupposetheinfostationhasKdatapacketstobedistributedtonmobilenodes.Forthisscenario,inthecentralsquarelettheelementaryinteractionisfortheinfostationtosendtwopacketstoamobilenode,whileinregularsquareletstheelementaryinteractionispacketexchangesbetweentwomobilenodes. 140

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TheeffectivenessofincentivemechanismIMismeasuredbytheexpectedpacketdeliverytime,i.e.,TD(IM).Forbroadcastscenarios,letTIMdenotethetimeforallmobilenodestoobtainKdatapacketsunderthemechanismIM,thenTD(IM)=E[TIM] K.ThesmallerthetimeTD(IM),themoreeffectivethemechanismIM.ThetimeTD(IM)fordifferentmechanismIMischaracterizedbythefollowingTheorem: Theorem5.1. [TD(IM)forBroadcastingKDataPackets]Whenallnodesarefullycooperativeandnonetworkcodingschemeisused(i.e.,IM=C),thenTD(C)=)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(logn+n K.Whenallnodesarenon-cooperative,and(1)ifbarteringisused(i.e.,IM=B),thenTD(B)=)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(log2n+n K;(2)ifvirtualcurrencyisused(i.e.,IM=VC),thenTD(VC)=)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(logn+n K;(3)ifourC4isusedwithk=Kandpisaconstant(i.e.,IM=C4),thenTD(C4)=)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(1+n K. Proof. Weobservethatforagiventimeslot,whetheramobilenodecanobtainanewpacketdependsontwoconditions:(i)Whetherthisnodeisselectedintoacommunicationpair?ThisconditionischaracterizedbyLemma 8 andisindependentoftheincentivemechanismused.(ii)Whenthisnodeisselectedintoacommunicationpair,whetheritcanmakeasuccessfulexchangewithitspartner?Here,weassumethatforanarbitrarynodepair,thereexistsaprobabilitypEofsuccessfulexchangewhichonlydependsontheincentivemechanism.Wewantastatic,summarizedcharacterizationofpE,i.e.,weobtainavalueofpEwhichisaveragedoverallpossiblenodepairsandalltimeslots.Wethensimplifyouranalysisbyassumingthatgiventheincentivemechanism,theprobabilityofsuccessfulexchangeforanynodepairatanytimeslotisthesameasthisvalue.Itisnothardtoseethatthiskeyassumptionisinconsistentwiththeinteractiondynamics.Nevertheless,oursimulationresultsagreecloselywiththeanalyticalresults,indicatingthatthisassumptionworkswellinsystemswithmoderatelylargenumberofnodesn50.Basedonthisassumption,wecanmodelthedynamicsofobtainingpacketsforagivenmobilenodeasadiscretetimeMarkovchainillustratedinFigure 5-5 .Denotethe 141

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Figure5-5. Statetransitiondiagramforobtainingpacketsofamobilenode. stateasthenumberofpacketsremainingtobeobtainedbyamobilenode.InitiallyanodeisatstateK.Sincethersttwopacketsmustbeobtainedfromaninfostation,thenextstateisK)]TJ /F6 11.955 Tf 9.35 0 Td[(2.Subsequently,instatess2f1,,K)]TJ /F6 11.955 Tf 9.36 0 Td[(2g,therearethreepossibilitiesineachtimeslot: Withprobability,thenodeisintheactivecentralsquareletandobtainstwopacketsfromtheinfostation.Thestategoesfromstos)]TJ /F6 11.955 Tf 9.96 0 Td[(2. Withprobability,thenodeisinanactiveregularsquareletandobtainsonepacketfromanothermobilenode.Thestategoesfromstos)]TJ /F6 11.955 Tf 9.96 0 Td[(1. Withprobability1)]TJ /F7 11.955 Tf 10.61 0 Td[()]TJ /F7 11.955 Tf 10.62 0 Td[(nonewpacketsareobtainedbecausethenodeisnotselectedintocommunicationpairsorbecausethenodecannotmakeasuccessfulexchangewithitspartnerandthestatestaysthesame.For,becausetheinfostationalwayshastheincentivetosendpacketstomobilenodes,thisprobabilityisindependentoftheincentivemechanismandfromLemma 8 (2)weknowthat=pI=(1=n).For,obviously=pE=(pE).Wedenotetheexpectedrstpassagetimefromstateitostate0asPTi,where2iK)]TJ /F6 11.955 Tf 10.82 0 Td[(2.Conditioningonthenextstatetransitionandrearrangingyieldsthedifferenceequation,PTi=1 ++ +PTi)]TJ /F9 7.97 Tf 6.58 0 Td[(1+ +PTi)]TJ /F9 7.97 Tf 6.58 0 Td[(2wheretheboundaryconditionsaregivenbyPT0=0andPT1=1 +.Usingz-transforms,wecanobtainPTi=i(+2)+1)]TJ /F15 11.955 Tf 11.95 13.27 Td[()]TJ /F19 7.97 Tf 6.58 0 Td[( +i (+2)2 142

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ItisobviousthatE[TIM]=1=+PTK)]TJ /F9 7.97 Tf 6.59 0 Td[(2,where1=istheexpectedtimeuntilanoderstobtainstwopacketsfromtheinfostation.Whenn!1,wecanobtainE[TIM]!K +1 pI=K pE+1 pI.Whenallmobilenodesarefullycooperativeandnonetworkcodingtechniquesisused,pEistheprobabilitythatanodehasanewpacketforanothernode.Usingtheresultaboutcouponcollectingproblem[ 33 ],wehavepE=(1=logn).Therefore,TD(C)=)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(logn+n K.Whenmobilenodesarenon-cooperative,wehavethreepossibilities:(1)Forbartering,pEistheprobabilitythattwonodescanprovideanewpacketforeachother,thuspE=)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(1=log2nandTD(B)=)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(log2n+n K.(2)ForVC,weassumeeachnodehasenoughVC.ThereforeTD(VC)isthesameasTD(C)incooperativecase.(3)ForourC4,usingthepropertyofRLNC[ 67 ],wehavepE(1)]TJ /F6 11.955 Tf 12.3 0 Td[(1=q)2=(1).Therefore,TD(C4)=)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(1+n K. ThecostefciencyofincentivemechanismIMismeasuredbytheexpectednumberofcontrolpackets(i.e.,CP(IM))thatneedbetransmittedforobtainingonedatapacket.Forbroadcastscenarios,letCIMdenotetheexpectedtotalnumberofcontrolpacketstransmittedforonemobilenodetoobtainKdatapacketsunderthemechanismIM,thenCP(IM)=E[CIM] Kn.ThesmallerthenumberCP(IM),themorecost-efcientthemechanismIM.ThenumberCP(IM)fordifferentIMischaracterizedbythefollowingTheorem: Theorem5.2. [CP(IM)forBroadcastingKDataPackets](1)CP(C)=CP(B)=CP(C4)=0and(2)CP(VC)=)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F9 7.97 Tf 13.15 5.03 Td[(logn n.(Theproofisomittedforthereasonofspace.)InFigure 5-6 ,wecomparetheoreticalresultsobtainedfromTheorem 5.1 and 5.2 withsimulationresultswhenL=4.WeshowTD(IM)andCP(IM)averagedover100simulationrunsfordifferentIMs.Thenumberofnodesisheldconstantatn=50(m=25)forFigure 5-6 (a)andatn=100(m=64)forFigure 5-6 (b),whilethenumberofdata 143

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Figure5-6. TD(IM)andCP(IM)asfunctionsofK. packetsKisvaried.Figure 5-6 (c)showsCP(VC)forn=50,100.TheresultsaboutCP(C),CP(B),andCP(C4)areallzeros,andarenotshowninthegure.Inallcases,thedifferencesbetweenthesimulationresultsandourtheoreticalresultsarefoundtobeverysmall.FromTheorem 5.1 and 5.2 andsimulationresults,wecanmakethefollowingobservationsaboutbroadcastscenarios:ForallpossiblevaluesofK,thepacketdeliverytimerequiredbyourC4isalwayssmallerthanthatrequiredbyotherpossibleincentivemechanisms.OurC4isevenbetterthanthecasewhenallnodesarecooperativebutnocodingisused.Alltheseimprovementsareobtainedwithzerocost.Therefore,ourC4isthebestchoiceforbroadcasttrafcs.Weassumethenumberofdatapacketswhichneedbebroadcastedisprettylarge,i.e.,K=(n).Inthisregime,fromTheorem 5.1 ,wehaveTD(B)=)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(log2n,TD(VC)=(logn),andTD(C4)=(1).TheimprovementsconvergetoafactorontheorderoflognwhichisindependentofK.Itisbelievedinpreviouswork[ 19 51 115 142 167 ]thatforbroadcasttrafc,barteringisbetterthanvirtual-currencybasedschemes.Ourresultsalsoprovideananalyticalevidencetosupportthisbeliefbecausecomparedtobartering,theVCbased 144

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schemeonlyprovidesan(logn)improvementtothepacketdeliverytime,withacostof(n)controlpacketspernode.Wethenconsidermulticastscenarios.Thefollowingcorollaryshowsthatthesameperformancecanbeachieved. Corollary5.1. SupposetheinfostationhasKdatapacketstosendtonrandomlyselectedmobilenodes.ThevaluesofTD(IM)andCP(IM)remainonthesameorderasthatinTheorem 5.1 and 5.2 whenisaconstant. 5.4.3PerformanceAnalysisforPureUnicastTrafcsBasedonabovediscussions,weconcludethatwhenbroadcastormulticasttrafcsexistinthenetwork,allofthemshouldbeutilizedtogenerateVCCpacketsbyperformingintra-sessionnetworkcoding.VCCpacketsgeneratedinthiswaycanfacilitatepacket/serviceexchangeswithoutincurringanycost.However,thesituationwillbecomplicatedforscenarioswithpureunicasts.Inthissubsection,weassumethereexistatleastnunicatsessions,eachofthemisdestinedtoonemobilenode.Wedividetheseunicastsessionsintoggroups(1gn),eachgroup(calledcodinggroup)containingn=gdistinctsessions.OnlythedatapacketsdestinedtomobilenodesinthesamegroupwillbemixedtogenerateVCCpackets,i.e.,weperformgroup-basedinter-sessionnetworkcoding(withgroupingparameterg)togenerateVCCpackets.Theremayexistotherunicasttrafcs,buttheywillnotbeinvolvedingeneratingVCCpackets.Forpureunicastscenarios,TD(IM)istheexpecteddeliverytimeforunicastingonedatapacketfromtheinfostationtoarandomlyselectedmobilenode(orinareversedirection)undertheincentivemechanismIM,andCP(IM)istheexpectedtotalnumberofcontrolpacketsneededtosupportachievingTD(IM)forthatdatapacket.WhenamobilenodereceivesavalidVCCpacket,inbroadcast(ormixedtrafc)scenariosthisVCCpacketincursnocontroloverhead.However,inpureunicastscenarios,thisVCCpacketonlycontainsg=nusefulinformationonaverage.Therefore, 145

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thisVCCpacketincurs(1)]TJ /F3 11.955 Tf 12.22 0 Td[(g=n)controloverhead,i.e.,thereisnoVCCpacketwithoutincurringanycost.Thisisthekeyfeatureemerginginpureunicastscenarios.Anotheruniquefeatureforpureunicastscenariosisaboutrouting.FromLemma 8 (3),weknowthatthenetworkisdisconnected,andbecauseeverynodeperformsarandomwalk,thereisnowaytopredictcontacts.Therefore,thesourceordestinationnodecannotprovideincentivestointermediatenodesexceptthedirectnexthop.Asaconsequence,wecanonlyprovideincentivestosustaintwo-hoprelay[ 58 171 ],i.e.,everypacketcanatmosttaketwohopsfromthesourcetothedestination.InSection 5.5 ,wewillremovethisconstraintbyconsideringhumanmobilitytracesfromtherealworld.Werstcharacterizeperformanceoftraditionalapproaches: Theorem5.3. [PerformanceofBarteringandVCforPureUnicasts]Forpureunicasttrafcs,(1)TD(B)=(n)andCP(B)=0;and(2)TD(VC)=)]TJ /F2 11.955 Tf 5.47 -.82 Td[(p nandCP(VC)=)]TJ /F2 11.955 Tf 5.48 -.83 Td[(p n.(Theproofisomittedhereforthereasonofspace.)Ithasalreadybeenshownin[ 171 ]thatwhenallnodesarefullycooperative,TD(C)=)]TJ /F2 11.955 Tf 5.48 -.82 Td[(p nforunicasts.FromTheorem 5.3 (2),wecanseethatVCbasedschemecanachievethelowerboundofTD(C)withlargecost.Thisisbecauseinordertoobtain)]TJ /F2 11.955 Tf 5.48 -.83 Td[(p npacketdeliverydelay,ittakes)]TJ /F2 11.955 Tf 5.48 -.83 Td[(p nnodestoprovidepacketrelayservices.Forbartering,becausethereisnomutualwantsbetweentwomobilenodesatall,theonlywayforamobilenodetoreceiveapacketistocontacttheinfostation,whichcausesalargedelayontheorderofn.ForourC4,bytakingdifferentvaluesofg,i.e.,thegroupingparameter,C4infactcanprovideaseriesofincentivemechanismswithdifferentnetworkperformanceasfollows. Theorem5.4. [PerformanceofOurC4forPureUnicasts]Forpureunicasttrafcs,(1)wheng=O)]TJ /F2 11.955 Tf 5.48 -.82 Td[(p n,TD(C4)=)]TJ /F2 11.955 Tf 5.48 -.82 Td[(p nandCP(C4)=p n)]TJ /F4 7.97 Tf 16.59 5.03 Td[(g p n;(2)wheng=)]TJ /F2 11.955 Tf 5.48 -.82 Td[(p n,TD(C4)=(g)andCP(C4)=n g)]TJ /F6 11.955 Tf 11.96 0 Td[(1. 146

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Proof. (1)wheng=O)]TJ /F2 11.955 Tf 5.47 -.83 Td[(p n,fromourpreviouswork[ 171 ]weknowthatasking)]TJ /F2 11.955 Tf 5.48 -.83 Td[(p nmobilenodestoactasrelaynodesisenoughtoachievethelowerboundofpacketdeliverydelayontheorderofp n.AskingmorenodestohelponlyincreasesthecostofourC4andcannotfurtherdecreasethepacketdeliverydelay.Therefore,wheng=O)]TJ /F2 11.955 Tf 5.48 -.83 Td[(p n,theinfostationonlyrandomlyselects)]TJ /F2 11.955 Tf 5.48 -.83 Td[(p nmobilenodesfromthesamegrouptoactasrelaynodes.Fromourpreviouswork[ 171 ],weknowthatinthissituationTD(C4)=)]TJ /F2 11.955 Tf 5.48 -.82 Td[(p n.Becauseeachcodedpacketonlycontainsg=nusefulinformationforaparticularmobilenode,eachcodedpacketincurs)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F4 7.97 Tf 13.15 5.03 Td[(g ncontroloverhead.Onaverage,eachmobilenodeneedexchangeonecodedpacketwitheveryrelaynode,andtotally)]TJ /F2 11.955 Tf 5.48 -.83 Td[(p ncodedpacketsneedbeexchangedforthedestinationnodesuccessfullydecodingonedatapacket.Therefore,CP(C4)=\0001)]TJ /F4 7.97 Tf 13.15 5.03 Td[(g np n.(2)wheng=)]TJ /F2 11.955 Tf 5.48 -.82 Td[(p n,theinfostationutilizesn=gmobilenodes(i.e.,allnodesinthesamegroup)toactasrelaynodes.FromtheproofofTheorem 5.1 weknowthatinordertosuccessfullydecodeoneoriginaldatapacket,onaveragethedestinationnodeneedwait(g)timeslots,andthisisexactlytheTD(C4).Becauseeachcodedpacketonlycontainsg=nusefulinformationforaparticularmobilenode,eachcodedpacketintroduces)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F4 7.97 Tf 13.16 5.03 Td[(g ncontroloverhead.Onaverage,eachmobilenodeneedexchangeonecodedpacketwitheveryrelaynode,andtotally(n=g)codedpacketsneedbeexchangedforthedestinationnodesuccessfullydecodingonedatapacket.Therefore,CP(C4)=)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F4 7.97 Tf 13.15 5.03 Td[(g nn g. FromTheorem 5.4 ,wedirectlyobtainthefollowingcorollary. Corollary5.2. Theeffectiveness-costtradeoffofourC4forpureunicastswhenCP(C4)=O)]TJ /F2 11.955 Tf 5.48 -.82 Td[(p nisgivenbyTD(C4)=n CP(C4)+1.Figure 5-7 illustrateaboveeffectiveness-costtradeoffofourC4forpureunicasts.Thesolidlinesherearetheoreticalresultswhilepointsrepresent(CP(C4),TD(C4)) 147

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Figure5-7. Theeffectiveness-costtradeoffsofourC4forpureunicasts. pairsobtainedfromsimulations.Forbothn=50andn=100,wecanseethatsimulationresultsagreewithourtheoreticalresults,i.e.,theeffectiveness-costtradeoffindeedexists.FromTheorem 5.4 ,Corollary 5.2 andsimulationresults,wecanmakethefollowingobservationsaboutunicasttrafcs:WhenCP(C4)=O)]TJ /F2 11.955 Tf 5.48 -.82 Td[(p n,TD(C4)isastrictlydecreasingfunctionofCP(C4),asshowninCorollary 5.2 andFigure 5-7 .Thismeansthatthereexistsafundamentaltradeoffbetweenpacketdeliverydelayandcost.Ifwewanttodecreasepacketdeliverydelay(i.e.,increaseincentiveeffectiveness),thentheimplementationcostofC4mustbeincreased.Byadjustingthegroupingparameter,C4providesasystematicwaytosmoothlytradeincentiveeffectivenessforimplementationcost.InourC4'stradeoffcurve,wheng=n,weobtainCP(C4)=0andTD(C4)=(n).Thisisthesameas(CP(B),TD(B))pair(cf.Theorem 5.3 (1)).Wenotethatthisisnotcoincidentbecausewheng=n,nointer-sessionnetworkcodingisperformedandnovalidVCCpacketsgeneratedinC4.Soitisequivalenttobarteringcases.InourC4'stradeoffcurve,wheng=1,weobtainCP(C4)=)]TJ /F2 11.955 Tf 5.48 -.83 Td[(p nandTD(C4)=)]TJ /F2 11.955 Tf 5.48 -.83 Td[(p n.Thisisthesameas(CP(VC),TD(VC))pair(cf.Theorem 5.3 (2)).Wenotethatthisisalsonotcoincidentbecausewheng=1,everyVCCpacketonly 148

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Table5-4. Datasetproperties DatasetCambridgeMIT DeviceiMotePhoneNetworktypeBluetoothBluetoothDuration(days)11246Granularity(sec.)600300#ofdevices5497#ofcontacts10,87354,667Datasource www.haggleproject.org reality.media.mit.edu contains1=nusefulinformation,whichtendsto0whennisprettylarge.Therefore,inthiscase,VCCpacketisequivalenttoe-cashpacket. 5.5ImprovingOurC4'sPerformancewithSocialContactInformationInprevioussection,therearetwoproblemsleftforourC4forpureunicastscenarios,i.e.,(1)two-hoprelayconstraintsand(2)unavoidableeffectiveness-costtradeoffs.WenotethatthesetwoproblemsarenotcausedbyourC4,insteadtheyaretheconsequenceofoversimpliedandunrealisticmobilitymodelusedforperformanceanalysis.Inthissectionweaddressthesetwoproblemsbyconsideringuniquefeaturesinreal-worldusermobilitypatterns.ThissectionalsoservestovalidateourC4'sperformancewithhumanmobilitytraces.Inthissection,weusetwoexperimentaldatasetsgatheredbytheHaggleProject(referredtoasCambridge)andtheMITRealityMiningProject(referredtoasMIT).ThecharacteristicsofthesedatasetsaresummarizedinTable 5-4 .Werstconvertallthesereallifedataintosocialcontactgraphs,withdevices/mobileusersasnodesetsandcontactsbetweentwonodesasedgesets.Fromthesesocialcontactgraphs,wecaneasilyndthattheyareheterogeneousbothintermsofedgesandcommunitystructures.Notethatfromtherandomwalkmobilitymodel,wecanonlyobtainahomogeneouscontactgraph,i.e.,acompletegraphwithequaledges.Therearenolinkheterogeneityorcommunitystructuresbecauseeverynodehasthesameprobabilitytocontacteveryothernode.Inwhatfollows,weshowhowtoutilizethesetwokindsofheterogeneitiesinrealitytofacilitateourC4design.HerewedonotmeanourC4can 149

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workonlyiftheinformationaboutsocialcontactgraphisavailable.WeonlywanttoshowthatC4canevenworkbetterifthisinformationisavailable.Infact,allDTNroutingschemestrytocollectandutilizethisinformation[ 178 ],andourC4doesnotrequireanythingmore. 5.5.1InformationHighwayandMulti-hopRelayTherstkindofheterogeneityiscallededgeheterogeneity[ 88 ].AsshowninFigure 5-8 (a),edgesareannotatedwithoneormoretimesatwhichtwonodescontact.Wecanseethatnotalledgesinthesocialcontactgrapharewiththesameimportance.Forexample,weassumethattheinfostationupdatesitsstatusinformationeverytimeslot,andeverymobilenodeexchangesitsneweststatusinformationabouttheinfostationwithothersduringcontacts.Then,wecanobservefromFigure 5-8 (a)that,althoughnodeGcancontacttheinfostation,formostoftimeslots,nodeGobtainstheneweststatusinformationabouttheinfostationfromnodeH.So,giventheexistenceoftheinformationpropagatinghighwayG)]TJ /F1 11.955 Tf 9.3 0 Td[(H)]TJ /F1 11.955 Tf 9.3 0 Td[(infostation,edgeG)]TJ /F1 11.955 Tf 9.3 0 Td[(infostationcanbedeletedfromthesocialcontactgraph.Forthesamereason,edgeI)]TJ /F1 11.955 Tf 9.3 0 Td[(infostationandedgeG)]TJ /F1 11.955 Tf 9.3 0 Td[(Icanalsobedeleted.Bycontinuingthisdeletingprocedure,atlast,wewillobtainatreerootedattheinfostation.Thistreeiscalledtheinformationhighway,i.e.,thestructureoffastindirectpathsfromtheinfostationtoallmobilenodes[ 88 ].Byassumingcontactsbetweenpairsofnodesareperfectlyperiodic,wecancalculatethedelayoftheneweststatusupdatesfromtheinfostationtoeverymobilenodealongthistree.TheresultisFigure 5-8 (b),theinformationhighwaywitheverynodeknowsthelatencyvaluefromtheinfostationtoitself.Theconcentriccirclesdenoteballradiiincreasingby5minutes,andthedistanceofeachnodefromthecommoncenterisitslatencyvaluefromtheinfostation.Foreverynodeu,itslatencyvaluedT(u)isakindoftemporaldistancebetweenitselfandtheinfostation,andthereforecanbeusedasaroutingmetric.AllnodeshaveincentivestomaintainacorrectdT(u)forsecuringitsownpacketdeliveryand 150

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Figure5-8. Informationhighwayandmulti-hoprelay. Figure5-9. Thedistributionofhopcounts. theinformationhighwaycanbeconstructedinafullydistributedway.Forunicastsfrommobilenodestotheinfostation,multi-hoprelaycanbeperformedinthefollowingway.Whenarelaynodeucontactsanodev,andvhasasmallerdT(v),thenpacketsdestinedtotheinfostationwillberelayedtov.InFigure 5-8 (b),oneofsuchrelaypathfromnodeAtotheinfostationisillustratedbyredcurves.Theunicastsfromtheinfostationtothemobilenodeswillbeevensimpler.Theroutingpathwillfollowthetreeofinformationhighway,becausebythedenitionofinformationhighway,thisisthepathwiththeminimumpacketdeliverydelayfromtheinfostationtothedestination. 151

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TheonlyproblemlefthereiswhyourC4cansustaintwo-more-hoprelaywhentheinformationhighwayemerges.Forthoseintermediaterelaynodeswhodonotcontactwiththesourceorthedestinationofpacketpkt,theincentivesforthemtobuypktcomefromthefactthethenodeswhichareclosertothedestinationwillhavemoreincentivestobuyit.Therefore,theycansellitalittlelaterandfromthisbuy-sellproceduretheycanobtainmoreVCCpackets.Forexample,inFigure 5-8 (b),onthepathfromnodeAtotheinfostation,nodeEwillbuypktfromnodeDbecauseheknowssomenodeslikeEhasthewillingnesstobuypkt,becauseEisclosertotheinfostationandhasmorechancestosellpkttotheinfostation.Figure 5-9 showshowmanyhopsthepacketstaketoreachthedestination.Wecanseethatforbothdatasets,thehopcountdistributionsofourC4arethesameasthatofthecaseswhenallnodesarefullycooperative.Therefore,wecanconcludethatourC4providesadequateincentivestosustainmulti-hoprelaywhentheinformationhighwayisavailable. 5.5.2CommunityStructureandGroupingParameterSelectionThesecondkindofheterogeneityiscalledcommunitystructures.Socialcontactgraphstypicallycontainpartsinwhichthenodesaremorehighlyconnectedtoeachotherthantotherestofthegraph.Thesetofsuchnodesisusuallycalledacommunity[ 126 ].Figure 5-10 (a)illustratesfourcommunitiesinanexemplarysocialcontactgraph,eachwithadifferentcolor.Wecanobservethat(1)atypicalmemberinacommunityislinkedtomanyothermembers,butnotnecessarilytoallothermembersinthecommunity,(2)differentcommunitiesmayoverlap.InFigure 5-10 (a)overlappingpartsareemphasizedbygreycolor.TheuseofcommunitystructuresinourC4isstraightforward.Whenthereexistsnobroadcastormulticasttrafcs,weneedtogroupsomeunicastsessionsdestinedtodifferentnodestoperformgroupbasedinter-sessionnetworkcodingtogenerateVCCpackets.ThewayofgroupingwillsignicantlyaffectourC4'sperformanceas 152

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Figure5-10. Communitystructuresinthesocialcontactgraph. discussedinSection 5.4.3 .Weknowthatifwedecreasethegroupsize(i.e.,n=g)toreducecosts,theeffectivenessofC4willalsobereduced.Thisrelationshipiscalledeffectiveness-costtradeoffscharacterizedinCorollary 5.2 .However,alltheseresultsarebasedonthecontactgraphswithoutcommunitystructures.FortheexamplegiveninFigure 5-10 (a),wecandirectlyobservethat,whenwegrouptheyellowcommunityandbluecommunityseparately,comparedtogroupingyellowandbluecommunitiestogether,theeffectivenesswillnotbeaffected.Thereasonissimple:mostofnodesintheyellowcommunityinfacthavenochancetocontactmostofnodesinthebluecommunity.Therefore,whentheVCCpacketsareonlyvalidfortheyellowcommunity,thereisnoefciencyloss.So,everycommunitycanhasitsownvalidVCCpackets,justlikeeverycountryhasitsowncurrency.Comparedtoaglobalcurrency,thelossofefciencyisignorable.Therefore,whenweshrinkthegrouptothecommunity,wecanreduceoverheadwithoutlosingeffectiveness.Basedonabovediscussions,weconcludethatthebestchoiceofgroupingshouldbebasedoncommunitystructuresinthesocialcontactgraphs. 153

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Figure5-11. Theeffectiveness-costtradeoffsunderdifferentg(ork)values. Theproblemleftishowtoidentifycommunitiesautomatically.Thedifcultycomesfromtheuniquefeaturesofcommunitystricturesinsocialgraphs.Becauseanodeinacommunityisnotnecessarilylinkedtoallothernodesinthecommunity,thecommunityisnotaclique(i.e.,acompletesubgraph).Differentcommunitiesmayoverlap,andthereforetraditionaldivisiveandagglomerativemethodscannotbeapplied[ 126 ]forthissituation.Here,weuseatechniquecalledk-clique-communitiesproposedin[ 126 ].Afullyconnectedsubgraphofknodesiscalledak-clique.wedeneak-clique-communityastheunionofallk-cliquesthatcanbereachedfromeachotherthroughaseriesofadjacentk-cliques,wheretwok-cliquesaresaidtobeadjacentiftheysharek)]TJ /F6 11.955 Tf 11.32 0 Td[(1nodes.Figure 5-10 (a)illustratesoverlapping4-cliquecommunities,andFigure 5-10 (b)showstheprocedureofformingtheyellowcommunityinFigure 5-10 (a).Obviously,thebestchoiceofkherecorrespondstothebestchoiceofgroupingparameterg.Herewedevelopadistributedschemetondtheoptimalkandconstructthecorrespondingk-clique-communities.Wecallthemaximalcompletesubgraphsascliques.Incontrasttothek-cliques,cliquescannotbesubsetsoflargercliques,thereforetheyhavetobelocatedinadecreasingorderoftheirsize.Thelargestpossiblecliquesizeinthegraphisdeterminedfromthedegree-sequence.Therefore,theinfostationrstndsthenodewiththehighestdegreeinthesocialcontactgraph,thenndsthecliquecontainingthatnode.Afterrecordingthisclique,theinfostationdeletes 154

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thenodeanditsedgesinthiscliquefromthesocialcontactgraph.Thentheinfostationrepeatsthisprocedureuntilnonodesareleft.Afterthat,therecordedsetofcliquesisasetofindependentk-cliquesinoriginalsocialcontactgraph.Thentheinfostationchoosesseverallargestk-cliquesasseedgroupsandbroadcastthenodelistsforeachseedgroup.Othernodescanchoosegroupmembershipsbythemselves.Theinfostationwillperformgroup-basedinter-sessionnetworkcodingonpacketsfromthesamegrouptogenerateVCCpackets.Sinceallmobilenodeshavetheincentivestochoosethemostappropriategroupsforthemselves,afterseveraliterations,thebestcommunitystructureswillemergesbythemselvesandtheoptimalk(org)valuewillbeautomaticallydetermined.Figure 5-11 showsall(CP(C4),TD(C4))pairsobtainedbythesystemitself.Wecanobservethatthereindeedexistsanoptimalvalueofg,whichcorrespondstothecriticalpointatwhichwhenwefurtherincreasethecostCP(C4),thecorrespondingpacketdeliverydelayTD(C4)cannotbedecreased.ThisoptimalvalueforCambridgedatasetis6whilethatforMITdatasetis8.Thismeansthat,forexample,thecommunitystructuresofsocialcontactgraphofCambridgedatasetismostappropriatelydescribedby6-clique-communities. 5.6ChapterSummaryThebenetsofutilizingnetworkcodinginwirelessnetworks,suchasimprovingthroughput,reducingenergyconsumption,simplifyingtransmissionscheduling,etc.,havebeenextensivelystudiedintheliterature(cf.[ 33 67 171 ]andreferencestherein).However,asfarasweknow,ourC4isthersttodemonstrateanewpotentialbenetbroughtbynetworkcodingforwirelessnetworks,i.e.,providingincentivesforcooperation.Incentiveproblemsinwirelessnetworkswithnetworkcodinghavealsobeenstudiedin[ 28 176 ].Bothofthemapplygametheorytostudycodingstrategiesadoptedbyindividualnodesandstillusevirtualcurrency(credit)tocreateincentives.InourC4,weassumethatthemainapplicationofMWNsistoprovideInternetaccess 155

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services,andmosttrafcinaMWNwillgothroughtheinfostation,whichenforcesrandomlinearnetworkcoding(cf.Section 7.3.1 )toimprovethewholesystem'sefciency.Therefore,noroomisleftforself-interestedmobilenodestochangecodingstrategies,andcodedpacketscanbesafelyutilizedtocreateincentives.Bytreatingnetworkcodingasatooltoproducevirtualcommoditycurrency,ourC4providesaneffectiveandlightweightsolutiontoinducecooperation,whichisimpossiblefor[ 28 176 ]basedontraditionalvirtualcurrency.HowtoextendourC4topurewirelessadhocnetworkswillbeaninterestingandimportanttopicandwewillstudyitinourfuturework. 156

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CHAPTER6TRUST-BASEDROUTINGANDNON-CLASSICALROUTINGALGEBRA 6.1ChapterOverviewAwirelessadhocnetwork(WANET)isacollectionofwirelessmobilenodesdynamicallyformingatemporarynetworkwithoutrequiringanycentralizedauthorityorxednetworkinfrastructure[ 20 71 119 131 ].InWANETs,everynodehastworoles:itservesasbothanindependentend-system(terminal)andarouteractivelyparticipatinginpacketforwardingforothernodes.Communicationsbetweentwonodescanbeperformeddirectlyifthedestinationiswithinthesource'stransmissionrange,orthroughintermediatenodesactingasrouters(multi-hoptransmission)ifthedestinationisoutsidesource'stransmissionrange.IthasbeenwidelyrecognizedthatWANETsareamongthemostpromisingnetworkingtechnologiestoproviderapid,untetheredaccesstoinformationandcomputing,eliminatingthebarriersofdistance,time,andlocationformanyapplicationsrangingfromcollaborativeanddistributedmobilecomputingtodisasterrecovery(suchasre,oodandearthquake),lawenforcement(crowdcontrol,searchandrescue)andmilitarycommunications(command,control,surveillanceandreconnaissance)[ 119 131 ].However,duetotheiropen,distributedanddynamicnature,WANETsarehighlyvulnerabletovarious(externalorinternal)maliciousattacks[ 20 71 ]andselshbehaviorsofparticipatingnodes.AsroutesinWANETsarecomposedofonlymobileterminalsbasedonamulti-hopmechanism,WANETsmayencountersituationswhereintermediatenodesinterruptpacketdeliveryadverselyorselshly.Accordingly,itisveryimportanttorealizerobust,efcientandsecureroutingprotocolsinordertoguaranteealarge-scaledeploymentofWANETs[ 20 132 ].OneimportantobservationisthattrustplaysanuniqueandfundamentalroleinsolvingtheroutingsecurityproblemsofWANETs,andthereforetrustevaluation,trustpropagation,andtrustworthypathselectionshouldbeintegratedintoroutingprotocoldesigninordertoenhancetheroutingsecurityofWANETs[ 20 54 148 ]. 157

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Trustunderliesanysecuritymechanisms.Eachwirelessnodeactsasarouterandparticipatesintheroutingprotocol.Routingreliesthereforeonexplicitorimplicittrustrelationshipsamongparticipatingnodes.Onlybasedontheexistingtrustrelationships,cankeymanagementsbecorrectlyperformedandothercryptographicprimitivesbefurtherdevelopedforsecuringaWANET. Trustprovidesauniformmechanismtodefendmaliciousandselshbehaviors.Itiswellknownthatcryptographictechniquesalonecannotcopewithroutingdisruptionsduetointernalattacksandselshbehaviors.Anetwork-widetrust/reputationmanagementsystemcanhelpininternalattackerdetectionandencouragethedesiredcooperativebehaviorsamongindividualnodes.WhennodesjoinandleaveaWANETdynamically,trustalsoprovidesaquantitativewayforanodetojoinallothernodesintointernalandexternalcategoriessecurely. Trustextendsthedesignspaceofsecureroutingprotocols.Insteadofapplyingcryptographicprimitivestosecureexistingroutingprotocols,wecanactivelyutilizethetrustasametricforasourcenodetoselectthemosttrustworthypathtothedestination.Thisleadstothesocalledsecure-awarerouting,whichincorporatessecurityrequirementsintotheroutingoperationsfromtheverybeginning[ 20 148 ].Tofacilitatetheimplementationofthisidea,varioustrustmetrics,whichquantifytrustrelationshipsaccordingtodifferentapplications'securityrequirements,havebeendesignedandintegratedintoroutingmetrics1inthesecurityresearchcommunity.Forexample,PGP-styleauthenticationschemeswithcerticationchains[ 23 168 ]useabinarytrustvaluation(e.g.,1-or-0,all-or-none).Reputation-basedschemes[ 124 151 ]employrealnumberstomeasurethetrustworthiness.Insomeevidence-basedschemes[ 77 152 ],atwo-dimensionalvectorin[0,1]2describesthetrustopinion.In[ 166 ],trustmeasurementisevencombinedwithotherQoSrequirementstoactastheroutingmetric.Specictrustinferenceandtrustworthypathselectionalgorithmsarealsodesignedforpeer-to-peernetworks,adhocnetworks,andsensornetworks,whichprovideabundantchoicesfortrust-basedroutingprotocoldesign(cf.Section 5.2.2 fordetaileddiscussions). 1Whenthetrustmetricisutilizedasa(partof)routingmetric,wecallthelatterasthetrust-relatedroutingmetric,orjusttrustmetricforshort,andthecorrespondingroutingprotocolastrust-basedrouting. 158

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Whiletheseapplication-speciedtrustmetricscapturedifferentcharacteristicsoftheirtargetscenariosortrustrelationships,thereisalackofunderstandingontheimpactoftrustmetricsontheoperationsofroutingprotocols.OneimportantlessonwehavelearnedfromInternetroutingprotocoldesignisthatingeneralwecannotarbitrarilychangeoneroutingmetrictoanotherwithoutconsideringtheroutingprotocolsusedinthenetwork.Ifroutingmetricsareunscrupulouslycombinedwithanincompatibleroutingprotocol,theroutingprotocolmayfailtondanoptimalpathorleadtoroutingloops[ 143 145 ].Thisprincipleisstillapplicablehere,sincethetrustmetricisjustaspecialkindofroutingmetrics,androutingprotocolsinWANETssharemanycommonfeatureswiththeirInternetcounterparts[ 131 164 ].Infact,ourcasestudyshowsthatsometrustmetricsmentionedaboveleadtoroutinganomalieswhentheyarecombinedwithaDijkstra-basedroutingalgorithm(cf.Example4and5inSection 6.2 fordetaileddiscussions).Therefore,withoutagoodunderstandingontheinteractionsbetweentrustmetricsandroutingprotocols,theevaluationanddesignoftrust-basedroutingarestillattheempiricalstage.Inthenetworkingresearchcommunity,atheoreticalframeworkcalledroutingalgebra2hasbeendevelopedforthestudyofthecompatibilityofroutingmetricsandroutingprotocolsinthecontextofQoSrouting[ 143 ]andBGPprotocols[ 145 ]usedintheInternet.Ithasalsobeenextendedandappliedtomulti-hopwirelessnetworksrecently[ 108 164 ].Sowhydonotwejustusetheexistingroutingalgebrasmentionedabovetostudytrustmetrics(asapartortotalroutingmetrics)?Thekeypointhereisthattrustmetricsaresignicantlydifferentfromnormalroutingmetricssuchasthenumberofhops,dataratesorotherQoSrequirements.Inwhat 2Pleasenotethatthealgebraheremaynotmeetallconditionsinthedenitionofalgebrausedinthemathematicalliterature,ratheritrepresentsanabstractalgebraicstructure. 159

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Figure6-1. PhysicalgraphandtrustgraphforanexemplarypathfromAtoE.Here,solidlinesanddashedlinesrepresentwirelesscommunicationlinksandtrustrelationships,respectively.Wecanobservefromthisexamplethat:forcomputingthenormalroutingmetric(e.g.,datarate)ofpathA!B!C!D!E(A!Eforshort),i.e.rA!E,weonlyneed4linksinformationinphysicalgraph(b);whileforcomputingthetrust-basedroutingmetricofpathA!E,i.e.,tA!E,weneedatleast7linksinformationintrustgraph(c). follows,weidentifyveuniquefeaturesoftrustmetrics,whichmakepreviousresultsunapplicabletotrust-basedrouting.Firstofall,thetopologicalstructuresrelatedtotrustmetricsaremorecomplicated.Fornormalroutingmetrics,onephysicalgraph,whichconsistsofedgesrepresentingphysicallinksbetweenindividualnodes,isenoughtodescribethetopologicalstructureoftheroutingproblem.Fortrustmetrics,trustrelationshipsamongindividualnodesformanothertopologicalstructurecalledtrustgraph,whichmayhavetotallydifferentstructurescomparedwiththephysicalgraph(refertoFigure 6-1 foranillustration).Trust-basedroutingisrestrictedbythephysicalconditionsaswellastrustconditions,andthereforerequiresanewroutingalgebrabuiltuponbothgraphs.Secondly,trustmetricshavedifferentalgebraicpropertiescomparedwithnormalroutingmetrics.Forexample,inExample3inSection 6.2 ,wewillshowthattrustmetricsingeneralarenon-distributive,whilemostofpreviousroutingalgebrasassumethatdistributivityholds[ 27 57 108 143 145 164 ].Sincethispropertyplaysakeyroleintheanalysisofnormalroutingmetrics,relatedresultscannotbeappliedherewhereitisnotsatised.Thirdly,fortraditionalroutingmetrics,themetricvalueofonephysicallinkisindependentofthatofotherlinks.Forexample,thebandwidth(ordelay)ofonelinkwillnotbeaffectedbythebandwidthofanother 160

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link,ifthesetwolinksdonotinterferewitheachother.However,trustcanbepassed(propagated)betweendifferentusers,whichmeansthetrust-basedroutingmetricsofdifferentphysicallinksaredependent.Obviously,thisdependencywillcomplicatetheanalysisoftrust-basedrouting.Fourthly,differentgroupsofpeoplemayhavedifferentrulestoestablishandhandletrust,andtherefore,trustmetricsaregroupdependentandnon-uniform.Whenanend-to-endcommunicationrunsacrossmultiplegroups,moreeffortneedstobemadetomodeltheinter-operationbetweendifferenttrust-basedroutingprotocols.Lastly,byitsverynature,trustisacomplexandmultidimensionalphenomenon[ 54 ].Somekindsoftrustlikereputationareuniversallyacceptableandtransitive(socialtrust),whileotherkindsoftrustisjustapersonalopinion(personaltrust),whichisnon-transitiveandincomparableatall.Thebreadthofthemeaningoftrustexcludesanypossibilitytomodeltrustmetricswithasinglealgebraicstructure.Multipleonesareneededtomodeldifferentkindsoftrust(i.e.,heterogeneityoftrust)andtheircombinationtooneapplicableroutingmetricisanewtopic.Tosumup,thediversityandcomplexityoftrustmetricsrequireasystematicanalysisoftheirpropertiesandthecorrespondingrelationshipswithroutingprotocols.Basedonanon-classicalalgebraicstructurecalledbi-monoid,inthischapterwewilldevelopaformaltheorytoinvestigatethecorrectness,optimality,andinter-operativityoftrust-basedroutingprotocolsforWANETs.Therestofthischapterisorganizedasfollows.InSection 6.2 ,werstutilizesomesimpliedexamplestoprovideourmotivationfordevelopingtheformalapproachadoptedinthischapter.Then,wedevelopanabstractframeworktofacilitateourstudyontrust-basedroutinginSection 6.3 .Next,wedevelopanon-classicalpathalgebrabasedonbi-monoidtostudyindirecttrustinferenceproblemsinSection 6.4 .Afterthat,inSection 6.5 weprovideasystematicanalysisoftherelationshipbetweentrustmetricsandtrust-basedroutingprotocolsbyidentifyingthebasicalgebraicpropertiesthatatrustmetricmusthaveinordertoworkcorrectlyandoptimallywithdifferentgeneralizeddistance-vectororlink-staterouting 161

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Figure6-2. Diversityoftrustmetrics. protocolsinWANETs.InSection 6.5 weextendourframeworktomodeltheinteractionsbetweendifferenttrust-basedroutingprotocols,andcharacterizetheconditionsunderwhichthecorrectnessandoptimalityofroutingoperationscanbeguaranteedintheWANETswheremultipleroutingprotocolscoexistordifferenttrustmetricsareadopted.Finally,weconcludethischapterinSection 6.7 6.2MotivatingExamples:WhyDoWeNeedaFormalStudy?Inthissection,weutilizesomesimpliedexamplestoprovideourmotivationfordevelopingthisformalapproach.Example1:diversityoftrustmetrics.Considerareputationsystembasedondirectinteractions.Anode'sexperiencewithanothernodeismodeledasabinaryevent:positiveornegative.Evidencehr,siisconceptualizedintermsofthenumbersofpositives(r)andnegatives(s).Basedonthissimpleevidencespace,varioustrustmetricshavebeenproposedintheliterature,accordingtodifferentdesignrationale: atrustvalueintherealinterval[)]TJ /F6 11.955 Tf 9.3 0 Td[(1,1](Figure 6-2 (a));[ 151 ] apairht,ciin[0,1]2,wheretandcrepresenttrustandcondencevalues(Figure 6-2 (b));[ 152 ] atriplehb,d,uiin[0,1]3,whereb,dandurepresentbelief,disbeliefanduncertaintyvalues,respectively,andb+d+u=1(Figure 6-2 (c));[ 77 ] atrustvaluecanbecombinedwithotherQoSrequirements(likedatarateanddelay)toformacombinedroutingmetric(Figure 6-2 (d)).[ 166 ] 162

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Infact,reputationisjustonepossibleinterpretationoftrust.Indifferentscenarios,trustmayhavetotallydifferentmeaningsandaccordinglyitshouldbeevaluatedindifferentways[ 20 54 148 ].ConsideringthediversityofemergingWANETsandtheirapplications,itwouldbefoolishtoattempttodeneacommontrustmetric.Therefore,moreandmoretrustmetricswithdifferentformsandinterpretationswillemerge.Diversityoftrustmetricsraisestwoproblemsfortrust-basedrouting:(1)ItispossiblethatmultipletrustmetricsareadoptedinonesingleWANET,thereforeweneedagenerallyapplicableframeworktofacilitateourstudyonthecompatibilityofdifferenttrustmetrics.(2)Ifwedonotwanttoexaminetrustmetrics(whichmayhaveinnitepossibilities)onebyone,wemustndawaytoabstractallthesemetricsandidentifythekeypropertiesrelatedtothecorrectnessandoptimalityofroutingprotocols.Example2:diversityofoperationsontrustmetrics.Consideranindirecttrustinferenceprobleminareputationsystem.Whentwonodeshaveneverdirectlyinteractedwitheachotherbefore,thereisnodirecttrusttobeevaluatedforthem.However,whenthesetwonodesmeet(i.e.,theyareineachother'stransmissionrange),theystillneedtoinferanindirecttrustbetweenthembasedonexistingdirecttrustsinthenetworks.TakeFigure 6-3 (a)asanexample,wherethenumberoneachlinkrepresentsthetrustvalueandwewanttoinfertheindirecttrustvalueit(v1,v0)fromv1tov0basedondirecttrustvalues.Therealsoexistvariouswaysintheliteraturetofullthistask(referto[ 152 ]and[ 54 ]andthereferencestherein): Wecouldchoosethestrongestpath,determinedbythepathwiththehighestminimumtrustvalue,andtakethelowesttrustvalueonthatpathasit(v1,v0).Basedonthisinferencealgorithm,thestrongestpathishv1,v3,v5,v0iwithit(v1,v0)=0.7. Wecouldchoosethestrongestpath,determinedbythepathwiththehighestproductofalltrustvaluesonthepath,andtaketheproductofalltrustvaluesonthatpathasit(v1,v0).Basedonthisinferencealgorithm,thestrongestpathishv1,v2,v4,v0iwithit(v1,v0)=0.49. 163

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Figure6-3. Algebraicpathformulationforindirecttrustinferenceproblems.In(a),solidlinesanddashedlinesrepresentdirectandindirecttrustrelationships,respectively.In(b),solidcurvesrepresenttrustpathsconsistsofdirecttrustrelationshipsonly. it(v1,v0)canbecalculatedastheweightedaverageoftheminimaofthetrustvaluesalongthedisjointpaths.Theweightsintheaveragingprocessaregivenbyv1'strustinitsdirectout-neighbors.Accordingtothisinferencealgorithm,it(v1,v0)=0.66.Everytrustinferencealgorithmmentionedabovehasitsownprosandcons.Here,wearenotinterestedinjudgingtheirusefulnessaccordingtodifferentapplicationscenarios,instead,wearefocusingondevelopinganabstractframeworksothatwecanreasontheirbehaviorasawhole.Fortrustinferenceproblem,weknowthatwecanapplyamathematicaltoolcalledpathalgebra[ 25 55 118 ].Wecandeneanoperatortoconcatenatetrustmetricsalongapath,thenweintroduceanotheroperatortoaggregatetrustmetricsacrosspaths(refertoFigure 6-3 (b)foranillustration).Whenandsatisfycertainproperties,usingpathalgebra,theproblemofcalculatingit(v1,vk)canbeformulateduniformlyastheonegiveninFigure 6-3 (b).FromFigure 6-3 (c),weobservethatalltrustinferencealgorithmmentionedabovecanbeincludedinthisformulationwithdifferentinterpretationsofand.Thisexamplesuggeststhat,insteadofexaminingeverypossibletrustmetricsandoperationsonebyone,wecanstudytheirbehaviorsasawholebydevelopinganalgebraicformalismthatabstractstrustrelationshipsandalsolinkresourcesaslabels,andmodelstheroutingoperationsascertainoperatorsonthelabels.Whatrelatedtothecorrectnessandoptimalityofroutingprotocolsarenotthecontents(themeaningof 164

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Figure6-4. Distributivityoftrustmetrics.Here,solidlinesanddashedlinesrepresentdirectandindirecttrustrelationships,respectively.Inbothcases,weassumethatp1=l1l2andp2=l3l4. thesetrustmetrics),buttheiralgebraicproperties.Therefore,weneeddescribe,classifyandanalyzedifferenttrustmetricsandoperatorsbasedontheiralgebraicstructures.Althoughinthisexampleweindicatethatpathalgebracanbeutilizedtoabstractandanalyzetrustinferenceproblems,weshouldalsopointoutthattraditionalpathalgebracannotbedirectlyappliedhere.Thereasonisthattrustmetricsmaypossesssomealgebraicpropertieswhichdonotholdintraditionalpathalgebra.Wewillillustratethispointinthefollowingexample.Example3:distributivityoftrustmetrics.ConsidertwotrustinferenceproblemsgiveninFigure 6-4 (a)and(b).UsingpathalgebradevelopedinExample2,indirecttrustvaluefromAtoEinFigure 6-4 (a)and(b)canbeexpressedasit(a)(A,F)=(p1p2)l5andit(b)(A,F)=(p1l5)(p2l5),respectively.Fortraditionalpathalgebrabasedonsemirings,distributesover,andthereforeit(a)(A,F)=it(b)(A,F).Thequestioniswhetherthesetwoexpressionsareequalfortrustmetrics?Weanswerthisquestionbyexaminingthephysicalmeaningsoftwoexpressions,respectively.ForFigure 6-4 (a),Fhasonlyonein-neighbor,i.e.D,whichmeansthatonlyDhasdirectinteractions(directtrust)withFbefore.Therefore,DwillactaswitnesstoprovidereportaboutF'strustworthinessbasedonitsownexperience,whichismeasuredbyl5.NodeAwillobtaintworeportsfromtwoindependentpaths,withbothreportsclaimingthatthetrustworthinessofFisl5.Thetrustworthinessfortherstreport 165

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Figure6-5. Examplesofroutinganomaliesintrust-basedrouting.Here,solidlinesanddashedlinesrepresentphysicallinksandphysicalpaths,respectively. itself(frompathA!B!D)isp1,whileforthesecondreport(frompathA!C!D)isp2.ForFigure 6-4 (b),AalsoobtainstworeportsstatingthetrustworthinessofFisl5,andthetrustworthinessforreportsthemselvesarep1andp2,respectively.ItseemsthatfromtheviewpointofA,thereisnodifferencebetweentwosituations.Unfortunately,thisisnottrue.Onekeyobservationisthat,inFigure 6-4 (a),thesetworeportsareissuedbythesamenodesD,whileinFigure 6-4 (b)thesetworeportsareissuedbytwodifferentnodesDandEseparately.Therefore,althoughnodeAcollectsthesamesecond-handevidence(tworeports)onFintwosituations,inFigure 6-4 (a)alltheseevidencescomefromasinglesourceDwhileinFigure 6-4 (b)theycomefromindependentsourcesDandE.Therefore,FismoretrustworthyforAinFigure 6-4 (b).Thisexampleshowsthatdistributivitydoesnotholdfortrustmetricsandthereforeweneeddevelopanewpathalgebrafortrustinferenceproblems.Inthenexttwoexamples,weshowthatroutinganomalies(suboptimalpathselectionandforwardingloop)mayariseeveninverysimplenetworksettingsintrust-basedroutingforWANETs.Theexistenceofroutinganomaliesnecessitatestheformalproofsofthecorrectnessandoptimalityoftrust-basedroutingprotocols.Example4:suboptimumoftrust-basedrouting.ConsidertheroutingproblemdescribedinFigure 6-5 (a).Letroutingmetricbeoftheformht,di,wheretistrustvalueanddisdelay.Themetricforaphysicalpathpishtp,dpi,wheretpistheminimaltrustvalueofphysicallinksalongthepathanddpisthedelayadditionalongthepath. 166

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Order4isdenedasalexicographicorder:ht1,d1i4ht2,d2i,t1>t2or(t1=t2andd1d2),wheretrustworthypathsarepreferred,withsmalldelaysbreakingties.Wesaythatpathp1=ht1,d1iisbetterthanorequaltop2=ht2,d2iifht1,d1i4ht2,d2i.Thissimplecombinedtrust-relatedroutingmetricisproposedin[ 166 ]andalsoappearsinFigure 6-2 (d)inExample1.Hereweareonlyinterestedintheroutingproblemwithv0asthedestinationinFigure 6-5 (a).Fornodev2,pathl1p1isnotbetterthanl1p2,becausel1p1=h0.5,11i,l1p2=h0.5,6iandh0.5,6i4h0.5,11i.Therefore,foranoptimalroutingprotocol,theselectedpathfromv2tov0shouldbel1p2.However,nodev1willalwaysselectpathp1forthedestinationv0becausepathp1isoptimalforit.Asaconsequence,foranyhop-by-hoproutingprotocols,theactuallyselectedpathfromv2tov0isl1p1,whichisnotoptimalforthesourcenodev2.Example5:forwardingloopintrust-basedrouting.ConsidertheroutingproblemdescribedinFigure 6-5 (b).AllpossiblephysicallinksaredescribedinFigure 6-5 (b)-i,andwetakenodev0asthedestinationnode.Therearethreetypesoftrustrelationships:friends,strangersandenemies,Figure 6-5 (b)-iidescribesalltrustrelationshipsinamatrixform.Basedonaboveinformation,foreachnodevi,wecanrankallpossiblepathsfromvitov0withorderrelation.Figure 6-5 (b)-iiigivespathrankingfornodesv1,v2andv3.Ourrankingisbasedontherelationshipsbetweenthesourcenodeandallintermediatenodesonthepath.Forlink-staterouting,themostpreferredpathscalculatedbyv1,v2andv3arel1p4,l2p6andl3p5,respectively.Therefore,thenext-hopforv1isv2,forv2isv3,andforv3isv1.Aloopv1!v2!v3!v1appears,andpacketswillbeforwardedinthisloopforever.Fordistance-vectorrouting,wetakev1asanexample.v1hasonlytwochoicesforthenext-hop:v2orv5.Intherstchoice,weassumethatv1takesv2asthenext-hop.Givenv1'schoice,v3willtakev6asthenext-hop.Givenv3'schoice,v2willtakev3asthenext-hop.Givenv2'schoice,v1shouldtakev5asthenext-hop,becausep5l1l2p6.Thiscontradictstoourassumptionthatv1takesv2asthenext-hop.Inthesecond 167

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Figure6-6. Systemmodelfor(a)trustmanagementinanycommunicationsystemsand(b)trust-basedrouting. choice,weassumethatv1takesv5asthenext-hop.Givenv1'schoice,v3willtakev1asthenext-hop.Givenv3'schoice,v2willtakev4asthenext-hop.Givenv2'schoice,v1shouldtakev2asthenext-hop,becausel1p4p5.Thiscontradictstoourassumptionthatv1takesv5asthenext-hop.Aswecanobservethatinbothchoices,weallendupwithcontradictions.Therefore,nodev1keepschangingitsroutingtable'sconguration.Thisiscalledrouteoscillation,anothernameforforwardingloopwithindistance-vectorrouting.Fromthesefewexamples,wecanobservethattrustmetricsaresignicantlydifferentfromtraditionalroutingmetricsandhencetraditionalroutingalgebracannotbeapplicable,whichcallsfornewtoolsforformalanalysis. 6.3AbstractFrameworkforTrustMetricsandTrust-BasedRouting 6.3.1SystemModelforTrustManagementIngeneral,trustmanagementinanycommunicationsystemcanbemodeledasaprocedurewiththreesequentialphases(refertoFigure 6-6 (a)foranillustration):Phase1:directtrustestablishment.Directtrustrelationshipscanbeobtainedfromtheevidencecreatedbythepreviousinteractionsorinheritedfromthepre-establishedsocialrelationshipsinthephysicalworld.Intheformercase,localmonitoringschemeslikeWatchdogandPathrater[ 111 ]andphysicalcontactscheme[ 24 ]havebeenproposedtocollecttherst-handtrustevidence.Inthelattercase,participatingnode 168

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onlyneedstoverifytheotherside'sidentity[ 168 ]basedoncryptographicoperations.Trustevaluationisamapping,whichtransformsthetrustevidenceorinherenttrustrelationshipsintodirecttrustmetrics.Phase2:indirecttrustinference.Whenthetrustrelationshipsare(atleastpartly)transitive3,moretrustrelationships(orindirecttrust)canbederivedfromthedirecttrust.Firstofall,directtrustinformationneedbepropagatedthroughoutthenetwork.Anyinformationgossipprotocolorbroadcastprotocolcanfulllthistask,andnospecicpathselectionschemeisneededhere.WealsoassumethatappropriatecryptographicprimitivesareavailableinWANETsinordertoprovideasecurepropagationofdirecttrustinformation.Then,basedonthesesecond-handtrustevidence,eachnodecanestablishnewindirecttrustrelationshipswithitsphysicalneighborsofwhichitknowsnothing.Trust-basedroutingisdifferentiatedfromtraditionalroutingbyintroducingtrustinferenceasoneofimportantpreprocessingforpathselectionandpacketforwarding,andthereforeshouldbeexplicitlyconsideredinourmodeling.Notethatwhendirecttrustrelationshipsareallnon-transitive,noindirecttrustcanbederivedfromphase2.Phase3:trust-basedoperations.Directandindirecttrustestablishedinprevioustwophaseswillbeutilizedtosupportallkindsofoperationswithsecurityrequirementsinthisphase.Fortrust-basedrouting,trust-relatedroutingmetricswillbeformed(e.g.,combinedwithotherQoSrequirements)andusedasthecriteriaforselectingmosttrustworthypathsbetweenanyS-Dpair.Phase1onlyconcernswiththeformingandevaluationofdirecttrustwithneighboringnodesandhasnodirectrelationswithroutingproblem,whichisorthogonalandcomplementarytotheresearchdescribedhereandthereforeexcludedfromtherestdiscussionsofthischapter.Basedonabovediscussions,wecanidentifytwo 3Wewilldenetransitiveandnon-transitivetrustinSection 6.3.3 169

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elementaryoperations,namely,trustinferenceandtrustworthypathselectionforanytrust-basedroutingprotocols,whichwillbeinvestigatedhere. 6.3.2GraphModelsforWANETsWeutilizegraphmodelstodescribephysicalresourcesandtrustrelationshipsinaWANET.Therefore,werstreviewsomebasicconceptsapplicabletoagraphmodel.ForalabeleddirectgraphG=(V,E,!),Visthevertexset,EVVisthelinkset,and!isafunctionwhichassignseachlinke2Ealabel!(e).Labelsherearetheabstractionofroutingmetricsortrustmetricsonlinks.Wealsoextendthisconcepttoapath,i.e.,thelabel!(p)ofapathp,whichisthemetricmeasuringthewholepath.Foredge(i,j)2E,wesaythatnodevjisanout-neighborofnodevi,andthatnodeviisanin-neighborofnodevj.Apathpfromv1tovnisdenotedbyp(v1,vn)=hv1,v2,,vniandp1,nforshort.Ifthelastnodeofpathpcoincideswiththerstnodeofpathq,thepqdenotesthepathformedbytheconcatenationofpandq.Apathissimpleorloop-freeifallnodesfromv1tovnaredistinct.Ifv1=vn,thenwesayp1,nformsaloop.ThephysicalresourcesofaWANETaremodeledbyphysicalgraphGH(V,EH,h)whereEHisthesetofdirectededgesrepresentingwirelesslinks4.ThedirecttrustrelationshipsaremodeledastrustgraphGT(V,ET,t)whereETisthesetofdirectededgesrepresentingdirecttrustrelationships.Whentrustistransitive,wecanderiveaugmentedtrustgraphGT(V,E,it)basedonGT(V,ET,t)andindirecttrustinferencescheme,whereE=HH.Foralink(i,j)2EH,theroutingmetricofthatlinkis 4Thiswirelesslinkcanbegeneralizedasanyphysicallinkwhichcanowpacketsbetweentwonodes.Forexample,inaWANETwithpartialinfrastructure,thislinkcanalsobeawiredlink.Inadelay-tolerantnetwork(DTN),thislinkcanbetwowirelesslinkscombinedwithonenodemovement. 170

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measuredbyafunctionr(i,j),i.e.,r(i,j),8><>:r(p(i,j),t(i,j))fornon-transitivetrustr(p(i,j),it(i,j))fortransitivetrust,whichconvertsthecombinationofphysicalpropertiesandtrustrelationshipintoatrustrelatedroutingmetrics.GraphGR(V,ER,r)iscalledroutinggraph,becausetrustworthypathselectionandpacketforwardingareactuallyperformedonit.Whentrustmetricsaredirectlyusedasroutingmetrics,r(i,j)=t(i,j).Notethat,eveninthiscase,trust/routingmetricsarestillconstrainedbyphysicalgraph,becauseER=EH.Duetothenodemobilityandlimitedcommunicationrangeofwirelesscommunicationtechniques,GT(V,ET,t)mayhavetotallydifferenttopologicalstructurefromthatofGH(V,EH,h).Notethatthispropertydistinguishesourstudyfrompreviousworkontrustinferenceortrust-basedroutinginP2Pnetworksoron-linesocialnetworks[ 54 ],wherethetrustgraphandthephysicalgraphareassumedtohavethesametopology. 6.3.3FormalizingTrustMetricSpaceBasedonabovediscussions,trustmetricsarejustlabelsonlinksoflabeleddirectgraphs.WedenotethenonemptysetofallpossiblelabelsasL.Inordertodenethespaceoftrust-relatedroutingmetrics,weneedintroducesomemathematicalstructuresonthesetL,butrememberthatwealsoneedminimizethenumberofspecicationsimposedonthestructureoverLinordertomaximizethegeneralityofourframework.Werstidentifytheindispensablestructuresfortrustmetrics.Firstofall,weshouldbeabletoderivethetrustmetricofapathfromtrustmetricsonindividuallinks.Therefore,weneedintroduceoperatorsovertheelementsofsetLtofacilitatethecombinationoflinklabels(ortrustmetrics).FromExample2inProjectDescription,wehavealreadyknownthatthefollowingtwocombinations(oroperations)overthelabelsareneeded:(1)Concatenationofseriallabelswithoperator;(2)Aggregationofparallellabelswithoperator.Atleastthesetwooperationsshouldbeclosed, 171

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i.e.,foralla,b2L,theresultsoftheoperationsabandabarealsoinL.Thismeansthatbycombiningtwolegallabelsweareguaranteedtoobtainalegallabeltoo.Obviously,thisisanalgebraicstructurebecausewedenesomeoperationsonasetandwerequiretheseoperationssatisfyingsomeconstraints/properties.Inthefollowingdiscussions,wewillintroducemorenecessarypropertiesofthesetwooperationsintheplacewheneverneeded,andbasedontheseoperator'sproperties,wecangetdifferentalgebraicstructureslikesemiringandbi-monoid(cf.Section 6.4 ).Wewillalsointroducenewoperators(likeinSection 6.5 )fortrustworthypathselections.Secondly,sinceweneedselectthemosttrustworthypathfromallphysicallyavailablepaths,weshouldbeabletocompareallelementsinL.Therefore,wealsoneedintroducesomekindoforderrelationintoL.Wedeneatotalpre-order4overLasfollows5.Fortwopathsorlinkspandqinagraph,let!(p)2Land!(q)2Ldenotetheirlabels.If!(p)4!(q),wesaythatpisweaklypreferredtoq(e.g.,pathpisatleastastrustworthyaspathq).If!(p)4!(q)and!(q)4!(p),thenwewrite!(p)!(q)andsaythatpandqareequallypreferred.Foranypair!(p),!(q)2Lwehaveeither!(p)4!(q)or!(q)4!(p).Functionsmaxandminaredenedwithrespectto4.Notethat!(p)!(q)mean!(p)4!(q)and!(p)6!(q),i.e.,strictlypreferred.Therefore,wecandenethenonemptylabelsetwithalgebraicstructureandorderrelationdescribedaboveastheabstracttrustmetricspace.Wedistinguishtwokindsoftrustinthisproject:transitiveandnon-transitivetrust.Fortransitivetrust,ifnodevihastrustt(i,j)innodevj,andvjhastrustt(j,k)innodevk,thenvishouldhavesometrustt(i,k)invkthatisafunctionoft(i,j)andt(j,k).Fornon-transitivetrust,however,wecannotobtainanyconclusionont(i,k),givent(i,j)andt(j,k).Example5inProjectDescriptionillustratesthisconcept.Wenotethattransitivetrust(andconsequently,indirecttrust)isimportantforany 5Recallthatapre-orderisareexiveandtransitiverelation. 172

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trust-basedroutingtobepracticalinlarge-scaledynamicWANETs.Becauseofthelargenumberofnodesandnodemobility,itisimpossibleforonenodetohavedirecttrustwithanyofitsphysicalneighbors.Whenthephysicallinkbetweentwonodesisneededtoperformmulti-hopcommunications,thetrustonthislinkmustbederivedbeforehand.Wealsonotethattrustisnotalwaystransitiveinreallife,andthereforewecannotexcludenon-transitivetrustfromourframework.Inthisproject,wewillrststudythesituationwhenalltrustsinaWANETaretransitive,whichiscalledhomogeneoustrustenvironment,andthenextendourstudytothesituationwhentransitiveandnon-transitivetrustscoexistinaWANET,whichiscalledheterogeneoustrustenvironment.WealsodistinguishtwosituationswhenthewholeWANETismodeledbyonetrustmetricspace(andaccordinglywithonetrust-basedroutingprotocol)orbymultipletrustmetricspaces(andaccordinglywithmultipletrust-basedroutingprotocols),whicharecalleduniformtrustenvironmentanddiversetrustenvironment,respectively.Wewillrststudythesimplersituation,i.e.,uniformtrustenvironmentinSection 6.4 andSection 6.5 .InSection 6.6 ,wewillstudytheinter-operationproblemsintroducedbymultipletrustmetricspaces. 6.3.4FormalizingRoutingProtocolsGiventheroutinggraph,foranyhop-by-hoproutingprotocol,itsmaintaskistogenerateandmaintainapathbetweeneachS-Dpairwiththedesirableproperties(denedbyroutingmetrics).Becauseforeachdestination,everysourcewillhaveinitsroutingtablethenexthoptoreachthatdestination,aspanningtreerootedateachdestinationisdenedimplicitlybythesetofroutingtablesresidinginanetwork.Wecallthisspanningtreeasin-treewiththerootnodeasthegivendestination.Therefore,anyroutingprotocolcanbeabstractedasthecollectionofrulesandprocedurestogenerateanin-treeforagivendestination. 173

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Figure6-7. Exemplaryin-treesforrootnodev0(destination). Weillustratethisconceptbythefollowingexample6.ConsideranetworkmodeledbytheroutinggraphG(E,EL,L)giveninFigure 6-7 (a),withthedestinationnodev0.Thisnetworkhasmanyin-treeswhoserootisv0;twoofthosein-trees,IT1andIT2,areshowninFigure 6-7 (b)and(c),respectively.Ineachofthesein-trees,thenodelabelxiofnodevicanbeexpressedusinganconcatenationoperatorasfollowsbasedonthestructuresandlinklabelsofthegivenin-tree. 1. Fortherootnodev0,x0=1(aspeciallabelwhichwillbeexplainedalittlelater). 2. Ifnodeviisnottherootandnodevjistheparentofnodeviinthein-treeandl(i,j)isthelabeloflink(i,j),thenxi=l(i,j)xj.Forexample,thelabelsofallnodesinin-treesIT1(cf.Figure 6-7 (b))andIT2(cf.Figure 6-7 (c))aregiveninTable 6-1 .Obviously,nodelabelisrelatedtothechoiceofin-trees,andtherefore,isrouting-protocoldependent.Nodelabelcanbeinterpretedinthefollowingtwoways: 1. Individually,labelxirepresentstheroutingtablecongurationfordestinationv0atnodevi.Forexample,inIT1,x2=l(2,4)x4meansthatatnodev2theout-linkforv0willbelink(2,4)withthenext-hopnodev4. 2. Givenothernodes'congurationsxj(j6=i),labelxialsorepresentsthechosenpathfromvitodestinationv0.Forexample,inIT1: 6Werestrictourselvestounicastsinthischapter.Ourdiscussionbelowisrestrictedtoonedestinationnodev0scenarios.However,itisstraightforwardtobeextendedtothecasewitharbitrarynumberofdestinationnodes(i.e.,onein-treeforeachdestinationnode). 174

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Table6-1. NodelabelsinFigure 6-7 ForIT1inFigure 6-7 (b)ForIT2inFigure 6-7 (c) x0=1x0=1x1=l(1,3)x3x1=l(1,3)x3x2=l(2,4)x4x2=l(2,1)x1x3=l(3,0)x0x3=l(3,0)x0x4=l(4,3)x3x4=l(4,0)x0 x2=l(2,4)x4=l(2,4)(l(4,3)x3)(givenx4)=l(2,4)(l(4,3)(l(3,0)1))(givenx3andx0),whichmeansthepathfromv2tov0chosenbyin-treeIT1willbep2,0=hv2,v4,v3,v0i.Here,x2in()formequalstopathlabelofp2,0,whichisdenotedasl(p2,0).Theexistinghop-by-hoproutingprotocolsinWANETscanbedividedintotwocategoriesaccordingtotheirdifferentpathcalculatingapproaches:namely,link-stateroutinganddistance-vectorroutingInthelink-stateapproachtorouting,eachnodebroadcastsupdatesofitslocaltopologyinformation(linkstate)totherestofthenetwork.Thesebroadcastscanbeperiodicorevent-driven.Byputtingtheupdatestogether,eachnodeisabletoreconstructtheroutinggraphGR(V,ER,r)fortheentirenetwork.GivenGR(V,ER,r),anodecanthenconstructitsroutingtablesappropriatelybyrunningDijkstra'sshortestpathalgorithm(whentakingthedistanceastheroutingmetric).Forthegeneralroutingmetrics,ageneralizedDijkstra'salgorithmisused.Giventhedestinationv0,eachnodeviwillcalculatethemostpreferredpathpi,0onGR(V,ER,r),accordingtor(pi,0)=minfr(pi,0)j8pi,02Pi,0g,wherePi,0isthesetofallpathsfromvitov0.Whenvjisthenext-hopnodeonpathpi,0,wehavexi=r(i,j)xjwherevj2Ni.ExemplaryroutingprotocolsforWANETsintheliteraturewhichfallinthiscategoryincludeLQSR,HSR,OLSRandHSLS[ 131 ].Inthedistance-vectorapproachtorouting,neighboringnodesexchange(advertise)vectorsofdistanceswitheachother.Eachentryinadistance-vectorcorrespondstoaparticulardestinationandcontainsthecurrentdistanceestimateoftheshortestpath 175

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fromthesourcetothecorrespondingdestination.Forthegeneralroutingmetrics,itmeansthateachnodevionlyknowsitsout-neighbors7Ni,itsout-goinglinks,anditsout-neighbors'nodelabels.Nodeviwillcalculateitsownnodelabelxi(whichhasone-to-onemappingwiththeentryinroutingtableforv0andtheselectedpathpi,0)usingageneralizedBellman-Fordalgorithm:xi=minfr(i,j)xjj8vj2Nig.ExemplaryroutingprotocolsforWANETsintheliteraturewhichfallinthiscategoryincludeAODVandDSDV[ 131 ].Basedontheabovediscussion,wecannaturallyintroduceourdenitionsonthecorrectnessandoptimalityofanyroutingprotocolRasfollows: Denition6.1. [R-Correctness]A(trust-based)routingprotocolRissaidtobecorrect,ifgivenanyGR(V,ER,r),xi's(i=1,2,,n)calculatedbyRformanin-tree. Denition6.2. [R-Optimality]A(trust-based)routingprotocolRissaidtobeoptimal,ifgivenanyGR(V,ER,r),xi's(i=1,2,,n)calculatedbyRsatisfy 1. xi'sformanin-treeand 2. 8vi2V:xi=minfr(pi,0)j8pi,02Pi,0g.Basedonthedenitionofin-trees,acorrectroutingprotocolRwillnotcreateanyroutingloops,i.e.,Ralwaysconvergestoloop-freestates.Foranoptimalroutingprotocol,foreverynodevi(i6=0),datapacketswillbeforwardedalongthemostpreferredpathpi,0amongallexistingpathsfromvitov0.Notethatinourroutingprotocolabstraction,somedetailssuchaswhentocalculatethenexthop(thedifferencebetweenreactiveandproactiverouting)areignored,sincetheyareirrelevanttothethemeofourstudyhere,i.e.,thecorrectnessandoptimalityofroutingprotocols.Alsoweexcludesourceroutingsincefortrust-basedroutingitis 7Weassumethatfordirectedlink(i,j),datapacketscanowfromvitovjwhilesignalingroutingpacketsmaybesentintheoppositedirection. 176

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impracticaltoassumerelaynodeswillforwardpacketsaccordingtothepathassignedbythesource. 6.4PathAlgebraforIndirectTrustInferenceInthissection,wedevelopanon-classicalpathalgebrabasedonbi-monoidtostudyindirecttrustinferenceproblems. 6.4.1AlgebraicFoundationsThegeneralframeworksusedinthisandnxetsectionsarebasedonthealgebraicstructureofsemigroupsandmonoids.Thus,werstbrieyreviewsomerelevantresults.8Asemigroup(S,)isanon-emptysetSwithabinaryoperatorsuchthat(foralla,b,c2S) ab2S(-Closure), (ab)c=a(bc)(-Associativity).Moreover,asemigroup(S,)iscalled(foralla,b2S) commutativewhenab=ba(-Commutativity), idempotentwhenaa=a(-Idempotency), selectivewhenab=aorb(-Selectivity).Notethataselectivesemigroupmustbeidempotent,buttheconverseisnottrue.Asemigroup(S,)mayhavesomespecialelements: "2Sisanidentityif8a2S:"a=a"=a, 2Sisanannihilatorif8a2S:a=a=, a)]TJ /F9 7.97 Tf 6.59 0 Td[(12Sisaninverseofa2S,ifaa)]TJ /F9 7.97 Tf 6.59 0 Td[(1=a)]TJ /F9 7.97 Tf 6.58 0 Td[(1a=". 8Referto[ 25 55 56 118 ]foramorecompletesurveyoftheissuesexposedhere. 177

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Notethat",anda)]TJ /F9 7.97 Tf 6.59 0 Td[(1denedaboveareuniqueiftheyexist.Asemigroupisamonoidifithasanidentity.Amonoidisagroupif8a2S:a)]TJ /F9 7.97 Tf 6.59 0 Td[(1exists.Abi-semigroupisanalgebraicstructure(S,,)withtwobinaryoperatorsandoverthesetSwhichsatisfy (ab)c=a(bc)(-Associativity), (ab)c=a(bc)(-Associativity),thatis,whenboth(S,)and(S,)aresemigroups.Foracommutativemonoid(S,),itisalwayspossibletointroduceapre-orderrelationoverS,denoted4,as:8a,b2S:a4b,9c2S:a=bc.Wecallthisrelationascanonicalpre-order,sincetheidentityandassociativityofensurethat4isindeedapre-order.Notethat-commutativityensuresthatthefollowingdenitionof4areequivalent:9c2S:a=bc,9c2S:a=cb.Thecanonicalpre-order4hasthefollowingproperties: If"(-identity)exists,then8a2S:a4". If(-annihilator)exists,then8a2S:4a. 6.4.2TrustInferenceProblemFormalizationAftertrustpropagationsubphase,everynodewillknowthetrustgraph.Ifthetrustistransitive,thefunctionalityoftrustinferenceistocalculatetheindirecttrustbasedonthetrustgraphwhichdescribesdirecttrustrelationships.Inthissubsection,werstformalizethisprocedureasfollows.GiventhetrustgraphGT(V,ET,t),wheret:ET7!Tisafunctionwhichassignseachedgee2ETalabelt(e)2T(i.e.,thedirecttrustmetricone),weneedtocompute 178

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theindirecttrustmetricit(vi,vj)forevery(vi,vj)2VV.Werstintroducetwooperationsoverthelabels: Concatenationofseriallabelswithoperator, Aggregationtwoparallellabelswithoperator.Mathematically,,:TT7!Taretwofunctionswhichcombinetwolabelsintoanewone.Therefore,thelabelofanontrivialpathp1,k=hv1,v2,,vki(i.e.,thetrustmetricofpathp1,k)ofGT,denotedt(p1,k),isgivenbyt(p1,k),(((t1,2t2,3)t3,4))tk)]TJ /F9 7.97 Tf 6.59 0 Td[(1,k,9andt(pi),1fortrivialpathpi=hvii.Asolutiontoanindirecttrustinferenceproblemisafunctionit:VV7!Tsuchthatit(vi,vi),1andforalli6=j,it(vi,vj),Lp2P(i,j)t(p),whereP(i,j)isthesetofallpathsfromvitovj.Inthischapter,wesolvethisproblemwithinthefollowingalgebraicstructure,calledbi-monoid: Denition6.3. [Bi-Monoid]GivenatrustgraphGT,wedenethenon-classicpathalgebraoverGTasanalgebraicstructure(T,,,0,1),where (T,,0)isacommutativemonoidwith0asitsidentity, (T,,1)isamonoidwith1asitsidentity, -identity0isalsoanannihilatorfor.Wecallthisalgebraicstructure(T,,)asbi-monoid.FromSection 6.4.1 ,weknowthatthereexistsacanonicalpre-order4overT.Weindicateitas4forshort. 9Wheniscommutative,itequalstot(p1,k),t1,2t2,3tk)]TJ /F9 7.97 Tf 6.58 0 Td[(1,k. 179

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Inpreviousworklike[ 152 ],trustinferenceproblemsaresolvedwithinthealgebraicstructurecalledsemiring: Denition6.4. [Semiring]Abi-monoid(T,,,0,1)isasemiringifdistributesover,i.e.,foralla,b,c2T: (ab)c=(ac)(bc)(LeftDistribution), c(ab)=(ca)(cb)(RightDistribution). 6.4.3VericationoftheBi-MonoidPropertiesHerewegivethephysicalmeaningsoftheconstraints/propertiesweimposeonthealgebraicstructureoftrustmetricsinDef. 6.3 andexplainwhythesepropertiesshouldbeheldforalltrustmetricsinpractice.Rememberthatweneedtokeepthepropertysetassmallaspossibleinordertoleavemoreroomfortrustmetricdesign.Theassociativityforbothandoperatorsallowstheincrementalcalculationoftrustmetrics:Ifonemorelabelneedstobeaggregatedintothecurrenttotal,thenitcanbedoneinonestep,withouthavingtorecallalllabelsthatwereaggregatedforthecurrenttotal.Thesamegoesforconcatenation.Commutativityforaggregationmakesirrelevanttheorderinwhichlabelsaretakenintoaccount(i.e.,whichoneisrst,whichoneissecond,etc.).The0element(identityfor,annihilatorfor)correspondstononexistenttrustrelationsbetweennodes.Therationaleisthatif0isencounteredalongapath,thenthewholepaththroughthisrelationshouldhavelabelequalto0.Also,suchpathsshouldbeignoredin-aggregation.The1element(identityfor)correspondstothemosttrustworthyrelationsbetweennodes.Thiscanalsobeseenasthetrustrelationofanodewithitself(i.e.,trivialpathinGT).Ifapathisextendedwithalinkoflabel1,thepathlabelshouldremainthesame.Thekeydifferencebetweenourworkandpreviousworkisthatourmodeldoesnotimposedistributivityontrustmetrics. 180

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Next,weintroduceanewpropertyoftrustmetrics,whichisnotincludedinourbi-monoidmodel(i.e.,Def. 6.3 ),butwebelieveitshouldholdinpractice,sinceanypathwithlabel1ismostpreferred. Denition6.5. [Absorptivity]AtrustgraphGT(V,ET,t)issaidtobeabsorptiveifforeverysimpledirectedcyclec=hv1,v2,,vk,v1iinGT,wehave14t(c).Absorptivegraphisageneralizationoftheabsenceofanegativecycleinagraphwithrealnumbersasedgeweights.Itplaysanimportantroletomakesurethesolutionoftrustinferenceproblemexists. 6.4.4SolvingPathAlgebraicProblems(1)SolvingtheProblemwithSemirings:Werstrecallhowtosolveatrustinferenceproblemwithintheframeworkofsemiring.TheoperationsandcanbeextendedintheusualwaytomatricesbuiltfromtheelementsofthesetT.LetMn(T)denotethesetofallnnmatricesoverT,andforA2Mn(T)letAijdenotethe(i,j)-entryofA.ForallA,B2Mn(T),wedeneABandABby(AB)ij,AijAijand(AB)ij,nLk=1(AikBkj).The-identitymatrix0and-identitymatrix1aregivenby:0ij,0and1ij,8><>:1ifi=j,0otherwise.Then(Mn(T),,)formanothersemiring.TheadjacencymatrixAofthetrustgraphGT(V,ET,t)isAij,8><>:t(i,j)if(i,j)2ET,0otherwise.DeneA(k)asA(k),1AAk,whereAk,AAA(ktimes).LetP(i,j),Pk(i,j)andP(k)(i,j)bethesetofallpathsinGTfromitoj,thesetofpathsfromitojwithlengthkandthesetofpathsfromitojwithlengthatmostk, 181

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respectively.Obviously,Pk(i,j)P(k)(i,j)P(i,j).Thefollowingconnectionbetweenmatrixpowersandpathsofacertainlengthiswellknown: )]TJ /F8 11.955 Tf 5.48 -9.69 Td[(Akij=Mp2Pk(i,j)t(p),(6)NotethattheproofofEq.( 6 )reliesonthe(left)distributionrule.FromEq.( 6 ),wedirectlyobtain )]TJ /F8 11.955 Tf 5.48 -9.68 Td[(A(k)ij=Mp2P(k)(i,j)t(p).(6)IfthetrustgraphGT(V,ET,t)isabsorptive,thenweonlyneedtoconsidersimplepaths(i.e.,norepetitionsofnodesalongthepathisallowed).InthegraphGTwithjVj=n,nopathlengthislargerthann)]TJ /F6 11.955 Tf 12.73 0 Td[(1,whichmeansthatP(n)]TJ /F9 7.97 Tf 6.58 0 Td[(1)(i,j)=P(i,j).ThereforeA(n)]TJ /F9 7.97 Tf 6.59 0 Td[(1)=A(n)]TJ /F9 7.97 Tf 6.59 0 Td[(1+k)foranyk0.FromEq.( 6 ),weobtainthesolution it(vi,vj)=Mp2P(i,j)t(p)=)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(A(n)]TJ /F9 7.97 Tf 6.58 0 Td[(1)ij.(6)ManyefcientalgorithmsareavailableintheliteraturetocomputeAk,see[ 25 55 56 118 ]andthereferencestherein.(2)EliminatingDistributivity:Ourwaytosolvethetrustinferenceproblemwithoutdistributivityisbasedonthegroupingfunctiongintroducedin[ 96 ].Thebasicideaisthatbyutilizingthegroupingfunctiong,werstconverttheprobleminabi-monoidBMintotheprobleminasemiringC.Then,wemappingbacklabelscomputedinCtolabelsinBM.LetM(T)bethesetofallcountablemultisetsthatarecomposedofelementsinT.Mathematicallythegroupingfunctiong:M(T)7!M(T)hasthefollowingproperties: 1. ForM1,M22M(T),letM1M2bethemultisetsuchthatM1M2,ft1t2jt12M1andt22M2g.Then,g(M1M2)=g(g(M1)g(M2)); 2. ForallMi2M(T),wherei2IandIisacountableindexset,wehavegSi2IMi=gSi2Ig(Mi); 3. IfM2M(T)then(M)=g(M). 182

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Intuitively,ifMisasetofpathlabels,thenggroupslabelstogetheraslongasthisdoesnotviolatethedistributivity.Property1)statesthatthegroupingprocessiscompatiblewith,i.e.,groupingbeforeatrustevaluationdoesnotchangetheresultoftheevaluation.Property2)requiresanaturalcommutativitypropertyofthegroupingprocess.Finally,Property3)statesthatthegroupingprocessiscompatiblewith.Basedonthegroupingfunctiong,wecanalwaysturnaBM=(T,,,0,1)intoC=(^T,^,^,0`,1`)asfollows: ^T,g(M(T)); 8^M1,^M22^T:^M1^^M2,g^M1^M2; 8^M2M(^T):^^M,gnS^Mo; 0`=g(0)and1`=g(1).Then,ithasbeenprovedin[ 96 ]thatthesolutioninBMis: it(vi,vj)=M0@^Mp2P(i,j)^t(p)1A(6)Notethatthecomputationof^L^t(p)inEq.( 6 )canbeperformedlikeinsemirings,i.e.Eq.( 6 ).ItiseasytoshowthatthetrustinferenceproblemhasasolutioninBMifandonlyifithasasolutioninCdenedaboveandGT(V,ET,^t)isabsorptiveifandonlyifGT(V,ET,t)isabsorptive.Therefore,theconditionthatthereexistsasolutionforBMisthatthetrustgraphGT(V,ET,t)isabsorptive.Giventhesolutionit(vi,vj),wewillhaveacompletegraphGT(V,E,it),whereE=TT,anditisafunctionwhichassignseachedgee2Eanlabelit(e)2T(i.e.,indirecttrustmetric).GraphGTiscalledaugmentedtrustgraph. 6.5RoutingAlgebraforUniformTrustEnvironmentInthissectionwedevelopanon-classicroutingalgebratostudythecorrectnessandoptimalityofroutingprotocolsundertheuniformtrustenvironment,i.e.,weassume 183

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allnodesinaWANETestablishandhandletrustinthesameway.WewillrelaxthisassumptioninSection 6.6 6.5.1Non-ClassicRoutingAlgebraforTrust-BasedRoutingInthephysicalgraphGH(V,EH,h),functionh:EH7!Hassignseachedge(i,j)2EHaphysicallabelh(i,j)2Hwhichdescribesthephysicalpropertiesofthatlink.Wedeneaspecialphysicallabel2H,whichcorrespondstonophysicallinkinGH.Aftertrustinferenceprocess,wecancombinethephysicalgraphwiththeaugmentedtrustgraphGT(V,E,it)toobtaintheroutinggraphGR(V,ER,r),whereER=EHandr:HT7!Risafunctionwhichassignseachedge(i,j)2ERalabelr(i,j)2Rwhichcombinesthephysicallabelandtrustlabelonthatlink,i.e.,r(i,j)=r(h(i,j),it(i,j)).Notethat,whentrustisnon-transitive,wedirectlyusetrustgraphstocombinewithphysicalgraphs,andobtainroutinggraphs,i.e.,inthiscaseweassumeGT=GT.GiventheaugmentedtrustgraphGT(V,E,it)androutinggraphGR(V,ER,r),themaintaskoftrust-basedroutingistondapathfromeachnodevi2V)-116(fv0gtothedestinationv0.Inwhatfollows,wedevelopanon-classicroutingalgebratoformalizethisprocedure. Denition6.6. [RoutingAlgebra]GiventheroutinggraphGR(V,ER,r),wedenethenon-classicroutingalgebraoverGRasanalgebraicstructureA=(R,,,0,1,),where (R,,0)isacommutativemonoidwith0asitsidentity, (R,,1)isamonoidwith1asitsidentity, -identity0isalsoanannihilatorfor. :RR7!RisafunctionwhichcombineslabelsontheowgraphofaphysicalpathinGRintoanewone.Beforeweproceed,somecommentsonDef. 6.6 seemsinorder. Donotconfuse0and1inAwiththeonesinBMofDef. 6.3 .Here0and1arespecialelementsinRnotinT.Alsonotethattheoperandsofandare 184

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Figure6-8. Flowgraphs,functionandtrustevaluationonpaths. elementsinR,andthereforeandheremayhavetotallydifferentmeaningscomparedwiththeircounterpartsinBM. Wecanalsointroduceacanonicalpre-order4overR,aswhatwehavedoneforBM.Alsoweindicateitas4forshortfromnowon.0isalsocalledtheleastpreferredlabelsince8a2R)-85(f0g:a0and1iscalledthemostpreferredlabelsince8a2R)-55(f1g:1a. Functioncanbeinterpretedasacombinedoperatorofand,anditsoperandisaspecialstructurecalledowgraph,whichwillbeexplainedindetailsasfollows.Foraphysicalpathpk,1=hvk,vk)]TJ /F9 7.97 Tf 6.59 0 Td[(1,,v2,v1iinGR,wedenetheowgraphFk,1ofthatpathasalabeleddirectedgraph(refertoFigure 6-8 (c)foranillustration)suchthat ThevertexsetofFk,1isV1,k=fv1,v2,,vkg. TheedgesetofFk,1isER(Fk,1)[E(Fk,1),where ER(Fk,1),f(vk,vk)]TJ /F9 7.97 Tf 5.17 0 Td[(1),(vk)]TJ /F9 7.97 Tf 5.18 0 Td[(1,vk)]TJ /F9 7.97 Tf 5.17 0 Td[(2),,(v2,v1)g(denotedbysolidlinesinFigure 6-8 (c)). E(Fk,1),f(vk,vk)]TJ /F9 7.97 Tf 5.18 0 Td[(2),(vk,vk)]TJ /F9 7.97 Tf 5.18 0 Td[(3),,(vk,v1)g(denotedbydashedlinesinFigure 6-8 (c)). ThelabeloneachedgeofFk,1isdenedby foredge(vi,vj)2ER(Fk,1),thelabelwillberi,j,i.e.,thesameasthatintheroutinggraph; 185

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foredge(vk,vi)2E(Fk,1)(i=1,,k)]TJ /F6 11.955 Tf 10.51 0 Td[(2),thelabelwillber(,it(vk,vi)),whichisthecombinationofnophysicallinkwiththeindirecttrustit(vk,vi)betweentwonodes.Obviously,ER(Fk,1)ERandE(Fk,1)E.FlowgraphFk,1forthephysicalpathpk,1includesallinformationthatcanbeutilizedtoevaluatethephysicalpathpk,1fromtheviewpointofthesourcenodevk.Itincludesthequalitiesofphysicallinkswhichconsistofpathpk,1aswellasthetrustrelationshipsofthesourcevkwithintermediatenodesonthatphysicalpath.Fortraditionalroutingmetrics(i.e.,h(i,j)inGH),apathmetriccanbesimplycalculatedfromthephysicallinkmetrics.Forexample,forpathpk,1=hvk,vk)]TJ /F9 7.97 Tf 6.59 0 Td[(1,,v2,v1iinGH,wehaveh(pk,1)=h(lk,k)]TJ /F9 7.97 Tf 6.59 0 Td[(1pk)]TJ /F9 7.97 Tf 6.58 0 Td[(1,1)=h(lk,k)]TJ /F9 7.97 Tf 6.59 0 Td[(1)h(pk)]TJ /F9 7.97 Tf 6.58 0 Td[(1,1),wherelk,k)]TJ /F9 7.97 Tf 6.58 0 Td[(1representslink(k,k)]TJ /F6 11.955 Tf 12.68 0 Td[(1).Fortrustrelatedmetrics,wecanobservefromFigure 6-8 thatingeneral,(pk,1)=(lk,k)]TJ /F9 7.97 Tf 6.58 0 Td[(1pk)]TJ /F9 7.97 Tf 6.58 0 Td[(1,1)6=(lk,k)]TJ /F9 7.97 Tf 6.59 0 Td[(1)(pk)]TJ /F9 7.97 Tf 6.59 0 Td[(1,1).However,wecanutilizeBMintroducedinSection 6.4 tocalculate(pk,1).NotethatforowgraphFk,1,(R,,,0,1)formsabi-monoidoverR.LetFk,1bethesetofallpathsfromvktov1inFk,1,(pk,1)canbecalculatedas(pk,1)=Lf2Fk,1f. 6.5.2ConditionsforCorrectandOptimalRoutingWeconsiderthefollowingpropertiesoffunction,whichplayanimportantroleinguaranteeingthecorrectnessandoptimalityoftrust-basedroutingprotocols.GivenaroutinggraphGR(V,ER,r),ifapathp(oralinkl)existsinGR,wewritep2GR(orl2GR)withaslightabuseofnotation.Wedene: 1. RoutingalgebraAisstrictly-left-monotonic,ifforalllinkl2GRandpathp2GR,iflp2GR,(l)6=0and(p)6=0,then(p)(lp). 2. RoutingalgebraAisstrictly-right-monotonic,ifforalllinkl2GRandpathp2GR,ifpl2GR,(l)6=0and(p)6=0,then(p)(pl). 3. RoutingalgebraAis-right-isotonic,ifforalllinkl2GRandpathsp,q2GR,iflp,lq2GRand(p)4(q),then(lp)4(lq).Aisstrictly-right-isotonicifwereplace4byinabovestatement. 186

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4. RoutingalgebraAis-left-isotonic,ifforalllinkl2GRandpathsp,q2GR,ifpl,ql2GRand(p)4(q),then(pl)4(ql).Aisstrictly-left-isotonicifwereplace4byinabovestatement.Basedonthepropertieswedenedforfunction,wecansummarizeourmainresultsasfollows: Theorem6.1. Alink-stateroutingprotocolisguaranteedtobecorrect,ifanonlyif(1)thetrustgraphisabsorptive;and(2)routingalgebraAis-right-monotonic,-right-isotonicandstrictly-left-isotonic.Alink-stateroutingprotocolisguaranteedtobeoptimal,ifandonlyif(1)thetrustgraphisabsorptive;and(2)routingalgebraAis-right-monotonic,-right-isotonicandstrictly-left-isotonic. Theorem6.2. Adistance-vectorroutingprotocolisguaranteedtobecorrect,ifanonlyif(1)thetrustgraphisabsorptive;and(2)routingalgebraAis-left-monotonic.Adistance-vectorroutingprotocolisguaranteedtobeoptimal,ifandonlyif(1)thetrustgraphisabsorptive;and(2)routingalgebraAis-left-isotonicand-left-monotonic.Thedetailedproofsoftheseresultsareomittedhere.Here,werstcompareourresultswithSobrinho'sclassicroutingalgebra[ 143 145 ].Althoughourresultsarealsocharacterizedbymonotonicityandisotonicityofaoperator(functionhere),ourfunctionistotallydifferentwithitscounterpartinSobrinho'sroutingalgebra(i.e.,).Thedifferenceistechnical.Ourfunctionisrelatedtomonoidendomorphisms[ 56 ],andtherefore,isnotincludedbySobrinho'sroutingalgebra,whichisbasedonorderedsemirings. 6.5.3IllustratingExamplesHere,weutilizesomeconcreteexamplestoexplainhowtouseourabstractresultsdescribedinprevioussubsection.WealreadygiveanexampleinExample4Section 6.2 wheretheroutingprotocolisnotoptimal.Itiseasytocheckthatitviolatestheisotonicproperty.Inwhatfollows,wewillutilizeExample5inProjectDescriptiontoinvestigatethecasewhichviolatesthemonotonicproperty. 187

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Figure6-9. Flowgraphsforthephysicalpathsfromnodev1tonodev0inExample5inSection 6.2 WerstexhibitthewaytoformalizethisproblemwiththeroutingalgebraproposedinDef. 6.6 .WemaketheitemsinAmoreconcreteasthefollowing:FortheroutingalgebraA=(R,,,0,1,),wedene R=ffr.,st.,en.,gwith0=en.and1=fr., fr.st.en., foralla,b2R,ab=awhenabbwhenba, foralla,b2R,ab=awhenbabwhenab, :RR7!RisafunctionwhichcombineslabelsontheowgraphofaphysicalpathinGRintoanewonewithoperatorsanddenedabove.Figure 6-9 showsowgraphsforthreephysicalpathsfromnodev1tothedestinationv0inExample5inProjectDescription.Here,weintroduceanewlabeltoindicatethelinkswhichwillnotaffectthecalculationofthetrustworthinessofapath.Theintuitionhereisthatthetrustworthinessofapathonlyinvolvesthetrustrelationshipsbetweenthesourcenodeandintermediatenodesonthepath,anddependsontheleasttrustworthyintermediatenode.Figure 6-10 givesanexampleofutilizingoperatorsanddenedabovetocalculatefunctionforthephysicalpathgiveninFigure 6-9 (c). 188

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Figure6-10. CalculatingfunctionforthephysicalpathgiveninFigure 6-9 (c). ItiseasytocheckthatthespecicationgivenaboveleadstothepathrankingdescribedinExample5inProjectDescription,whichagreeswithourintuitiononthetrustworthypaths.AlsoourformalizationguaranteesthatforeachowgraphF,(R,,,0,1)formsabi-monoidoverR.Therefore,wecanutilizeBMintroducedinSection 6.4 tocalculatefunctions.Notethat,forSobrinho'sclassicroutingalgebra[ 143 145 ],onlylabelsonphysicallinksareinvolvedintheformalization.Therefore,onlyhop-by-hoptrustrelationshipscanberemainedinSobrinho'sclassicroutingalgebra,whicharenotsufcientforthecalculationofthetrustworthinessofawholephysicalpath.Obviously,theintroductionofowgraphsinDef. 6.6 makesitpossibletoincludealltrustrelationshipsintheformalization.Basedontheseinformation,wecaneasilycalculatethetrustworthinessofawholephysicalpathwithfunctions.Next,withthehelpofthisframework,itiseasytocheckthatforbothlink-stateanddistance-vectorroutingapproaches,noin-treecanformandthereforethecorrectnessofroutingprotocolscannotbeguaranteed. 6.6RoutingAlgebraforGroup-BasedTrustEnvironment 6.6.1MotivatingExampleConsideradisasterrecoveryscenario,thelocalpoliceforcemayneedtocoordinatewithreghters,militaryforces,andmedicalcrewsbysharinginformationandcommunicatingwitheachotherregardlessoftheparticularnetworkingprotocolsthateachgroupuses.SeeFigure 6-11 foranillustration. 189

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Figure6-11. Trustinmultiplegroups. Obviously,differentgroups(indicatedbydifferentcolorsinFigure 6-11 )likepoliceforce,militaryforce,medicalcrewsetc.mayhavedifferentrulestoevaluateandhandletrust.Therefore,differenttrustmetricswillbeadoptedandcombinedwithdifferenttrust-basedroutingprotocols.Nownodev1asapolicewanttocommunicatewithnodev0,adoctor,throughthemosttrustworthypath.Multiplequestionswillnaturallyariseforthistask.Forexample,hownodev1evaluatethetrustworthinessofpathp1andp2,sincelinksalongeachpatharemeasuredbydifferenttrustmetrics?Howtocomparep1withp2?Whatkindsofoperationrulesthenodesconnectingdifferentgroups(likev3,v5,v9andv13inFigure 6-11 )needtofollowinordertomakesuretherewillbenoloopsinhop-by-hoprouting?Orunderwhatconditionsthepathformedbythenext-hopselectionsofallintermediatenodesisthemosttrustworthypathasdenedbythesourcenodev1?Allthesequestionscallfordevelopmentofauniedframeworktoenabletheanalysisofend-to-endcommunicationsoverheterogeneousWANETsgovernedbydistincttrustmetrics.Inthissection,weconsideragroup-basedtrustenvironment,wheremultiplegroupscoexistinaWANET.EachgroupicanbemodeledasoneroutingalgebraAi=(Ri,i,i,0i,1i,i)denedinSection 6.5 .Theproblemhereishowtoperformend-to-endtrust-basedroutingacrossmultiplegroups.Tobemorefocused,wemakethefollowingsimplicationshere: 1. Weassumeeachgroupisinahomogeneoustrustenvironment,i.e.,thereexistuniformedrulesforthedirecttrustestablishmentandindirecttrustinferencein 190

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eachgroup,andtheseproceduresareindependentamongdifferentgroups.Interactionsbetweendifferentgroupsonlyhappeninthepathselectionphase. 2. Weassumeallgroupsusethesamekindofroutingprotocols,eitherdistance-vectororlink-stateroutingprotocols.Theproblemsemergefromtheinter-operationsbetweentwokindsofprotocolsareuniversal,whichareirrelevanttotrustmetrics,i.e.,thethemeofthischapter.Inprevioussection,weuseroutingalgebratostudyhowtoguaranteecorrectroutingandoptimalpathselectioninonegroup.Ourkeyobservationhereisthattheproblemsinvolvedinmulti-groupcommunicationsareallrelatedtoconversionsoftrust-basedroutingmetricsbetweendifferentgroups.Therefore,basedonpreviousresults,theproblemsofcorrectandoptimalroutinginmulti-groupenvironmentcanbemorespecicallyreformulatedasthefollowing:whatconditionsmusttheconversionsbetweendifferentroutingalgebrasatisfyinordertoguaranteethecorrectandoptimalend-to-endtrust-basedrouting? 6.6.2ProblemFormalizationWerstformalizethenotionofconversionsbetweendifferentroutingalgebras. Denition6.7. [ConversionFunction]Givenk+1routingalgebrasAi=(Ri,i,i,0i,1i,i)fori=0,1,,kasdenedinDef. 6.6 .RoutingalgebraA0iscalledthehostalgebra,ifallpathsacrossmultiplegroupswillbemeasuredusingthelabelsinR0.Correspond-ingly,routingalgebraAj(j6=0)iscalledtheguestalgebra,sincethelabelofapathpingroupAjwillbeconvertedtothatinthehostalgebrabeforepathpcanbeutilizedbyothergroups.WedenetwoconversionfunctionsbetweenA0andAjas: 0!j:R07!Rj, j!0:Rj7!R0.Next,weshowhowtomodelbasicinter-operationsbetweenhostalgebra/groupA0andguestalgebraA1byutilizingconversionfunctions.Weonlyconsidertwogroupshere.Itisstraightforwardtoextendourdiscussiontomoregroups.RefertoFigure 6-12 foranillustration.Here,nodevmbelongstobothgroups,andactsasbridgerouter 191

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Figure6-12. Concatenationofpathsfromdifferentgroups. toconnecttwogroups.Nodevmndsapath,saypm,1,tothedesignationnodev1ingroupA1.Letthelabelcalculatedbyvmforpathpm,1be1(pm,1).Inthedistance-vectorrouting,nodevmneedstoadvertisetheinformationaboutpathpm,1toitsneighbor,sayvm+1ingroupA0.Thelabel1(pm,1)cannotbedirectlyusedbecauseforvm+1label1(pm,1)belongstodifferenttrustmetricsetR1,whichcannotbeunderstoodandfurtherprocessedbyvm+1.Therefore,vmneedrstconvert1(pm,1)into1!0(1(pm,1)),i.e.,thetrustmetricusedinR0.Notethat,weassumethetrustinferenceprocessisperformedineachgroupindependently.Therefore,nodevm+1hasnoindirectrelationshipswithnodesingroupA1exceptvm.Thelabelofpathpm+1,1=(vm+1,vm)pm,1canbesimplycalculatedas0(pm+1,1)=rm+1,m01!0(1(pm,1)),whererm+1,m2R0isthelabeloflink(vm+1,vm).Forlink-staterouting,sourcenodevkneedcalculatethelabel(i.e.,trustmetric)0(pk,1)2R0forthewholepathpk,1=pk,mpm,1.Thispathcanbeseparatedintotwoparts:subpathpk,mwithlabel0(pk,m)2R0andsubpathpm,1withlabel1(pm,1)2R1.Sincetwolabelsbelongtodifferenttrustmetricsets,theycannotbedirectlycomparedorcombined.Byutilizing1!0,wecansimplycompute0(pk,1)=0(pk,m)01!0(1(pm,1)).Twoimportantobservationscanbemadethroughthisexample.(1)AlthoughtherearethreeoperationsdenedineachroutingalgebraAi,i.e.,i,iandi,onlyiinvolvesinpathlabelcalculationsacrossmultiplegroups.Therefore,theinteractionsbetweenconversionfunctionsandioperatorswillplayakeyroletodeterminethepropertiesofthewholesystem(withmultiplegroups).(2)Ifweabstracteachpathcontainedinonegroup(likepathspk,mandpm,1inFigure 6-12 )asageneralizedlink, 192

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fromtheviewpointofgroupA0,conversionfunction1!0assignseachgeneralizedlinkfromothergroupsanew(generalized)linklabel.Theprinciplesestablishedforonegroupobviouslyapplyhere.Conversionfunctionscannotbearbitrary:inordertoguaranteethecorrectnessandoptimalityoftrust-basedroutingacrossmultiplegroups,someconstraintsonthemmustbeimposed.Tocharacterizethoseconstraintsiswhatwewilldonext. 6.6.3PropertiesofConversionFunctionsConversionfunctionpair(j!0,0!j)representsrelationshipsbetweenhostalgebraA0andguestalgebraAj.Herewerstconsiderthefollowingpropertiesofconversionfunctionpair,whichwillbeusedtoanalyzethecorrectnessandoptimalityofroutingacrossmultiplegroups.RecallthatforeveryroutingalgebraAi,acanonicalpre-order4icanbenaturallyintroducedfromi.Wedene:(1)Functionpair(j!0,0!j)isguest-order-preservedif 8ra,rb2R0,ifra40rbwehave0!j(ra)4j0!j(rb).(2)Functionpair(j!0,0!j)ishost-order-preservedif 8ra,rb2Rj,ifrajrbwehavej!0(ra)0j!0(rb). 8r2R0,r40j!0(0!j(r)).(3)Functionpair(j!0,0!j)distributesoverif j!0isbijectiveand0!j=)]TJ /F9 7.97 Tf 6.59 0 Td[(1j!0, 8ra,rb2Rj,j!0(rajrb)=j!0(ra)0j!0(rb). 8ra,rb2R0,0!j(ra0rb)=0!j(ra)j0!j(rb).Tofacilitateouranalysis,wedeneauniversalroutingalgebraA=(R,,,0,1,,4)uponA0andAj,where R=R0[R1[[Rk :RR7!Risafunctionsuchthat8a2Ri,b2Rj:ab,ajbifi=j,i!j(a)jbotherwise. 193

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Aswediscussedbefore,iandineverinvolveinpathcalculationsacrossmultiplegroups,whichmeansoperantesoforarealwaysthesame.Therefore,wedene: ab=aibifa,b2Ri, (p)=i(p)ifthewholepathpisingroupAi.ForA,4cannotbeintroducedfrom,andthereforeshouldbedenedindependently: 4isanorderrelationoverRsuchthat8a2Ri,b2Rj:a4b,a4jbifi=j,i!0(a)40j!0(b)otherwise.Given4,0and1isdenedasfollows: 02Rsuchthat8a2R)-55(f0g,a0, 12Rsuchthat8a2R)-55(f1g,1a.Basedonabovediscussions,wehavethefollowingresults: Lemma9. Orderrelation4denedinuniversalroutingalgebraAisatotalpre-orderoverRifallpairs(j!0,0!j)arehost-order-preserved.Proof:Arelation4onanonemptysetRis reexive,if8a2R:a4a; transitiveif8a,b,c2R:a4bandb4c)a4c; total,if8a,b2R:a4borb4a.Recallthatatotalpre-orderisareexive,transitiveandtotalrelation.Inwhatfollows,weprovethattheorderrelation4denedinuniversalroutingalgebraAhasthesethreeproperties,respectively.[Reexivity]Foranarbitrarya2R,becauseR=R0[R1[[Rk,thereexistsani2[0,k]suchthata2Ri.Bythedenitionof4,a4b,a4ibwhena,b2Ri.Recallthat4iisatotalpre-orderoverRiforalli2[0,k](i.e.,4iisreexive),therefore,a4aforalla2R. 194

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[Transitivity]Forarbitrarya,b,c2R,becauseR=R0[R1[[Rk,thereexistsi,j,l2[0,k]suchthata2Ri,b2Rjandc2Rl.Therearevedifferentcasesaccordingtotherelationshipsbetweeni,jandl.Here,wejustaddressthecasethati,j,larealldistinct.Othercasesaresimplerandcanbereasonedinasimilarway.Bythedenitionof4andhost-order-preservation,a4bmeansthati!0(a)40j!0(b)wheni6=jandb4cmeansthatj!0(b)40l!0(c)whenj6=l.Because40isatotalpre-orderoverR0(i.e.,40istransitive),weobtaini!0(a)40l!0(c),whichimpliesthata4cforalla,b,c2R.[Totality]Forarbitrarya,b2R,becauseR=R0[R1[[Rk,thereexistsi,j2[0,k]suchthata2Riandb2Rj.Wheni=j,since4iistotal,wehaveeithera4iborb4ia.Bythedenitionof4,thisimpliesthata4borb4a.Wheni6=j,since40istotalandrelatedconversionfunctionsarehost-order-preserved,wehaveeitheri!0(a)40j!0(b)orj!0(b)40i!0(a).Bythedenitionof4,thisimpliesthata4borb4a. 6.6.4ConditionsforCorrectandOptimalRoutingWerstanalyzedistance-vectorroutingprotocolsinmulti-groupenvironment: Theorem6.3. Givenk+1routingalgebrasAi(i=0,,k)andconversionfunctionpairs(j!0,0!j)(j=1,,k),adistance-vectorroutingprotocolRisguaranteedtobecorrectacrossmultiplegroupsif(1)Riscorrectineachroutingalgebra;and(2)allpairs(j!0,0!j)arehost-order-preserved.Adistance-vectorroutingprotocolRisguaran-teedtobeoptimalacrossmultiplegroupsif(1)Risoptimalineachroutingalgebra;and(2)allpairs(j!0,0!j)arehost-order-preservedandguest-order-preserved.Proof:[Part1:Correctness]ToprovethecorrectnessofRacrossmultiplegroups,thekeypointhereistoshowthatCondition(2)issufcienttopreservestrict-monotonicityacrossthealgebras.Weassumethatanodendapathpwithlabeli(p)2RifromalgebraAi,andthepathisextendedtoanlinklwithalabelrj(l)2Rj.Withtheassumptionthatallroutingalgebrasarestrictly-monotonicandCondition(2) 195

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holds,weneedprovetheextendedpathlphasastrictlylowerpreferencethantheinitialpath.Wheni=j,i.e.,theinitialpathisextendedintothesameroutingalgebraAi,theproofistrivial.Therefore,wefocusonthecasei6=j.SinceAjisstrictly-monotonic,j(p)jj(lp),whichcanbewroteas0!j(i!0(i(p)))jrj(l)j0!j(i!0(i(p))).Thenbythehost-order-preservingproperty,wehavej!0(0!j(i!0(i(p))))0j!0(rj(l)j0!j(i!0(i(p))))andi!0(i(p))40j!0(0!j(i!0(i(p)))).Bythedenitionof,weobtaini!0(i(p))0j!0(rj(l)i(p)).Finally,bydenitionof4,weobtaini(p)rj(l)i(p).[Part2:Optimality]WeneedshowthatCondition(2)guaranteesthepreservationofthe-right-isotonicitypropertyacrossthemultiplegroups,i.e.,fortwopathsp2Riandq2Rj,ifi(p)4j(q),thenforl2Rk,k(lp)4kk(lq).IntheuniversalalgebraA,thisisequivalenttok(l)i(p)4k(l)j(q).Therearevedifferentcasesaccordingtotherelationshipsbetweeni,jandk.Here,wejustaddressthecasethati,j,karealldistinct.Othercasesaresimplerandcanbereasonedinasimilarway.Fromi(p)4j(q),wehavei!0(i(p))40j!0(j(q)),therefore0!k(i!0(i(p)))4k0!k(j!0(j(q))).Forlinkl:k(l)k0!k(i!0(i(p)))4kk(l)k0!k(j!0(j(q))).Therefore,k(l)i(p)4k(l)j(q). Theorem6.4. Givenk+1routingalgebrasAi(i=0,,k)andconversionfunctionpairs(j!0,0!j)(j=1,,k),alink-stateroutingprotocolRisguaranteedtobecorrectandoptimalacrossmultiplegroupsif(1)Riscorrectandoptimalineachroutingalgebra;and(2)allpairs(j!0,0!j)arehost-order-preserved,guest-order-preservedanddistributive.Proof:BasedontheproofofTheorem 6.4 ,herewejustneedshowthatgivenonemorecondition,i.e.,distributivity,wecanguarantee(1)theisotonicitypropertyacrossthealgebrasand(2)theassociativityofinuniversalalgebraA.[Part1:Left-isotonicity]Wealreadyshowtheleft-isotonicityintheproofoftheorem 6.4 .Here,wejustneedproveforthreepathsp,q,lwherei(p)2Ri,j(q)2Rj 196

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andk(l)2Rk,ifi(p)4j(q),theni(p)k(l)4j(q)k(l).Wefocusonthecasewheni,j,karealldistinct.Fromi(p)4j(q),wehavei!0(i(p))40j!0(j(q)).FromA0'sleft-isotonicity,wehavei!0(i(p))0k!0(k(l))40j!0(j(q))0k!0(k(l)).Becauseconversionfunctionpairsaredistributive,weobtaini!0(i(p)i0!i(k!0(k(l))))40j!0(j(q)j0!j(k!0(k(l)))).Bythedenitionof,weobtaini(p)k(l)4j(q)k(l).[Part2:Associativityof]Forthreepathsp,q,lwherei(p)2Ri,j(q)2Rjandk(l)2Rk,wefocusonthecasewheni,j,karealldistinct.Wehave(i(p)j(q))k(l)=(i(p)i0!i(j!0(j(q))))k(l)=(i(p)i0!i(j!0(j(q))))i0!i(k!0(k(l)))=i(p)i(0!i(j!0(j(q))))i0!i(k!0(k(l))))i(p)(j(q)k(l))=i(p)(j(q)j0!j(k!0(k(l))))=i(p)i0!i(j!0((j(q)j0!j(k!0(k(l)))))=i(p)i(0!i(j!0(j(q))))i0!i(k!0(k(l))))Therefore,(i(p)j(q))k(l)=i(p)(j(q)k(l)). 6.7ChapterSummaryInthischapter,wedevelopaformalframeworkandtheorytoinvestigatethecorrectness,optimality,inter-operativityoftrust-basedroutingprotocolsforWANETs.Ourresultsobtainedherecanbeextendedintwoways.(1)Forindirecttrustinferenceproblems,weonlyconsiderthesituationwhenalltrustsinaWANETaretransitive.Whentransitiveandnon-transitivetrustcoexistinaWANET,anewalgebraicstructureforacombinedtrustmetricisneededandconsequentlynewalgorithmshouldbedesignedtoinferindirecttrustundernon-transitivetrustconstraints.(2)Fromrouting'spointofview,inourframeworkweonlyconsidertopology-basedroutingprotocols.We 197

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canextendourstudytolocation-basedroutinglikegeographicrouting,whichispopularforWANETs.Alsointhischapterwerestrictourselvestounicastrouting.Obviously,thetrustmetricsformulticast,broadcast,andanycastaretotallydifferentfromthatforunicast,andtheconceptofpathselectionwillbereplacedbytreeselection.Therefore,thesetopicsshouldbefurtherinvestigated. 198

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CHAPTER7SECURENETWORKPERFORMANCEOFLARGE-SCALEWANETS 7.1ChapterOverviewThegrowthofmoderncommunicationnetworks,suchastheInternetandwirelesscellularsystems,overthelastdecadehassurpassedmanyexpectations.Indeed,goingbackintimetotheoriginsofthesenetworks,itwouldhavebeenhardtoimaginetheimportanceandscaletowhichthesenetworkshavedeveloped.Now,projectingintothefuture,westronglybelievethatthistrendwillcontinue,ifnotaccelerate.Hence,thecommunicationdevicesandprotocolsoftodaymustbecapableofoperatingwiththesameefciencyintheverylarge-scalenetworksofthefuture.Thishighlightstheneedforasymptoticanalysisonanetworkanditscorrespondingprotocoldesign,whichcharacterizestheasymptoticbehaviorsofnetworkperformanceasitssizengrows.Thisisespeciallythecaseforwirelessadhocnetworks(WANETs)whichoffercommunicationsoverasharedwirelesschannelwithoutanypre-existinginfrastructure,sincemoreeffortneedstobemadetoharmonizethebehaviorofdifferentparticipantsandmanagedistributednetworkresourcestosupportend-to-end(e2e)communicationdemandscomparedtothenetworkwithinfrastructure.Obviously,thiskindofunavoidablecoordinationoverhead,whichmaybetolerableinsmall-scalenetworks,ispossibletobecomedominantfactorsinlarge-scalenetworksandshouldbequantitativelyanalyzed.Sinceboththroughputanddelayareimportantnetworkperformancemetrics,signicanteffortinthelastfewyearshasbeendevotedtounderstandingthescalinglawsonthroughputanddelayandtheirrelationshipinWANETs.Intheirseminalwork,GuptaandKumar[ 60 ]showthattheper-owthroughputcapacityforstaticWANETsscalesas(1=p nlogn)(refertoAppendixB forthestandardasymptoticnotationusedthroughoutthechapter)undertheassumptionthatnodeswithcommontransmissionrangearerandomlydistributed.Notethatthiswork[ 60 ]implicitlyusesauidmodel 199

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forestablishingthroughputscaling.LaterworkbyKulkarniandViswannath[ 90 ]consolidatestheresultin[ 60 ]withanexplicitconstant-packet-sizemodel.Followingthesamemethodology,thecorrespondingdelayof(p n=logn)andthecompletethroughput-delaytradeoffsofstaticWANETsarerstobtainedin[ 48 ].Recently,withthepercolationtheory,Franceschettietal.[ 46 ]showthattheper-owthroughputof(1=p n)isachievableifeachnodecanadjustitstransmissionrangethroughpowercontrol.Thethroughput-delaytradeoffsunderdifferentmobilitymodelsarealsostudiedintheliterature(e.g.,[ 11 48 50 100 101 120 140 154 ]).Adrawbackcommontoalltheaboveresultsistheneglectofsecurityrequirements,whicharereceivinggrowingattentioninrecentyearsbecausemanylarge-scaleWANETsareexpectedtobedeployedinhostilescenariossuchasmilitaryandhomelandsecurityoperations[ 177 ].Itisknownthatsecurityalwayscomeswithaprice,assecuringcommunicationsagainsttheadversarytypicallyconsumemorenetworkresourcesintermsofbandwidthand/orhardwarecapacities.Thispricemaybetolerableinsmall-scaleWANETs,butitmaydominatetheconsumptionofscarcenetworkresourcesinlarge-scaleWANETs.Thissituationmakestheinvestigationofthroughput-delaytradeoffswithsecurityrequirementsinlarge-scaleWANETsanimportantopenchallenge.AlthoughsecurityrequirementsinWANETsareapplication-dependent,inthischapterwefocusonthemostcriticalandfundamentalonewhichreectsthedistinctnatureofWANETsandenablesanalyticaltractability,thatis,werequirethatwirelesscommunicationsshouldoperateonsecurelinkswhenevernecessary.AWANETcanbeinformallyvisualizedasagroupofwirelesscommunicationdevices/nodes,heldbyuserscomingtogetherspontaneouslytoformanetworkforacommonpurpose(e.g.,emergencyresponse).Somekeyingmaterialsforprimarysecurityassociations(SAs),whichwewillformallydenelater(cf.Section 7.2.1 ),arealreadypre-conguredincommunicationdevicesbasedonthetrustrelationshipsamongthepersonsinvolved. 200

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TheproblemishowtoexploitthoseprimarySAstoprovidesecurecommunicationsforarbitrarynodepairswhenneeded.Neighborauthenticationorsecuringthephysicallink,whichprovideshop-by-hopsecurity,istherststepforprovidinge2esecurecommunicationsinallkindsofnetworks.ThisisespeciallycrucialforWANETssinceeverynodeneedstoactasaroutertoforwardpacketsforothers.Ifthenodecannotauthenticateitsneighbors1,howcanittrustthemtohandleitspacketcorrectly?Obviously,neighboringnodeswithprimarySAscanauthenticateeachotherdirectlywithpre-conguredkeyingmaterials,andthephysicallinksbetweenthemcanbesecuredaccordingly.SincethenumberandthedistributionofprimarySAsaredeterminedbytheembeddedsocialnetwork(e.g.,trustrelationship)ofusers,anodemaynothaveprimarySAswithanyofitsneighbors.Infact,theprobabilitythatanodesharesaprimarySAwithanyothernode,i.e.,pf,willbeverysmallinpracticewhenthenodepopulationnintheWANETincreases.Inthiscase,ifthephysicallinkstillneedbesecured,securelinkaugmentation(SLA)operations2arerequiredandperformedwiththehelpofphysicallyconnectedcommonfriendsoftwoendnodesofthisphysicallink.Whenpf<1,thereisingeneralnetworkperformancedegradationbecausewerequireallcommunicationsoperateonsecurelinks,andsomenetworkresources,i.e.,theunsecuredlinkscannotbeutilizedcomparedtoWANETswithoutsecurerequirements.AlthoughwecanobtainmorederivedsecurelinkswithSLA,network 1WesaytwonodesarefriendswhentheyhaveaprimarySA.WesaytwonodesareneighborsiftheirEuclideandistanceisnogreaterthanthetransmissionrangern.Thereisaphysicallinkbetweenanytwoneighboringnodes,andthislinkcanbesecuredwiththeprimarySAiftheneighboringnodesarealsofriends.Wecallthiskindofsecurelinksasprimarysecurelinks.Alinkcanalsobesecuredwiththehelpofotherauthenticatedneighbors;wecallthiskindofsecurelinksderivedsecurelinks.2SLAheremeanstheprocedureofsecuringaphysicallinkbetweentwoneighboringnodeswhicharenotfriends.AdetaileddescriptionofSLAoperationsisgiveninScheme2inSection 7.5.1 201

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resourcesconsumedbySLAisanotherkindofsecuritycostwhichshouldbetakenintoconsideration.Therefore,itisnaturaltoask:whatisthepriceofsecurity(performancedegradation)wehavetopayinWANETs?Canwedesignaprotocoltoachievetheoptimalsecurenetworkperformanceorminimizethepriceofsecurity?Inthischapter,weanswerthesequestionswithrigorousanalysisbasedonreasonableassumptionsonWANETs.Weformallycharacterizethetradeoffsbetweenkeypre-distributionrelatedtopfandsecurenetworkperformance.OurresultsshowthattheminimalpriceofsecuritywithSLAisstrictlysmallerthanthatwithoutSLA,whichtheoreticallynecessitatesSLAoperationsinWANETswithsecurityrequirements.WealsodesigntwoschemestoachievetheminimalpriceofsecuritywithorwithoutSLA,respectively.Furthermore,theseschemesprovideseveralimportantinsightsonprotocoldesignforsecurecommunicationsinWANETsasfollows:(1)Itisunnecessaryandevenharmfultothinkthatinordertoachievetheminimalpriceofsecurity,wehavetoobtainasmanyderivedsecurelinksaspossible.Infact,thephysicallinksneedbesecuredwithSLAarefewandshouldbecarefullyselected.(2)Ourschemesshowthatitispossibletoconstructthesecurebackboneandselectthephysicallinksthatneedbesecuredinatotallydistributedfashionwithnegligiblecommunicationoverhead,andthissecureinfrastructureisunrelatedtosource-destination(S-D)pairsandcanbereusedagainandagain.(3)Althoughingeneralsecurenetworkperformancedegradeswithpf,withorwithoutSLA,oneimportantexceptionwendisthatwhenpfis(1=logn),thesecurethroughputremainsattheGuptaandKumarboundof)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(1=p nlognpackets/timeslot,whereinnosecurityrequirementsareenforcedonWANETs.Thisimpliesthatevenwhenpfgoestozeroasthenetworksizebecomesarbitrarilylarge,itisstillpossibletobuildthroughput-order-optimalsecureWANETs,whichisofpracticalinterestsinceinmanypracticallarge-scaleWANETs,pfisverysmall. 202

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7.2BackgroundandRelatedWorkTheimpactofsecurityrequirementsontheperformanceofWANETsislargelyuntouchedintheliteraturewithonlyafewexceptions[ 14 168 ].Inthissection,werstreviewsomekeyingschemesandsecureoperationsrelatedtothefulllmentofoursecurityrequirementsandthenpresentrecentresultsonsecureconnectivityandthroughput,respectively.Wecomparetheseresultswithoursobtainedinthischapterandpointoutourowncontributions. 7.2.1OnPre-DistributionofKeyingMaterials/SAsWhenwesaytwonodeshaveaprimarySA,wemeanthattwonodestrusteachotherinthesensethateitherasymmetrickeyissharedbetweenthemortheyknoweachothers'authenticpublickeys.WefurtherassumethatSAsarealwayssymmetricbecausetrustrelationshipissymmetricinnature[ 39 168 ].TheconceptofSAhereiscloselyrelatedtothetopicofpre-distributedkeyestablishmentandmanagementinthesecurityprotocoldesign[ 23 26 38 71 ].Inwhatfollows,wesummarizesomerepresentativeschemesproposedintheliteratureanddemonstratethattheparameterpfisagoodabstractionoftrustrelationshipsamongnetworknodesregardlessoftheimplementationdetailsofkeyingschemes.EschenauerandGligor'sKeyPoolScheme[ 38 ]:Beforethenodesaredeployed,anoff-linetrustauthority(TA)willprovidealargekeypoolofsizeP.Eachnoderandomlypickskdifferentkeysfromthiskeypool.Therefore,twoneighboringnodeshaveaprimarySAiftheyshareatleastonecommonkeyinthekeypoolwithprobabilitypf,whichisgivenas pf=1)]TJ /F15 11.955 Tf 13.15 18.53 Td[()]TJ /F4 7.97 Tf 5.48 -4.38 Td[(P)]TJ /F4 7.97 Tf 6.58 0 Td[(kk )]TJ /F4 7.97 Tf 5.48 -4.38 Td[(Pk=1)]TJ /F6 11.955 Tf 16.3 7.92 Td[(((P)]TJ /F3 11.955 Tf 11.95 0 Td[(k)!)2 (P)]TJ /F6 11.955 Tf 11.96 0 Td[(2k)!P!,(7)wherethesecondequalityholdsforP>2k.Chan,PerrigandSong'sRandom-PairwiseKeyScheme[ 26 ]:Eachnodeidentity(ID)ispairedwithfotherrandomlyselecteddistinctnodeIDs,andapairwisekeyispre-generatedforeachpairofnodes.Thekeyisstoredinbothnodes'keyringalong 203

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withtheIDoftheothernodethatknowsthekey.Therefore,theprobabilitypfoftwoselectednodessharingaprimarySAisdirectlygivenby pf=f=n.(7)Hubaux,CapkunandButtyan'sSelf-OrganizedPublic-KeyScheme[ 23 71 ]:LikeinPGP[ 182 ],eachnode'spublicandprivatekeysarecreatedbythenodeitself.UnlikeinPGP,wherecerticatesofthepublickeysaremainlystoredincentralizedcerticaterepositories,certicatesinthisschemearestoredanddistributedbythenodesinafullyself-organizedmanner.Forsimplicity,weassumethateachnodehasffriendsandithasalreadystoredthecerticatesofitsfriends'publickeys.Therefore,theprobabilitypfthatanodecandirectlyauthenticateoneofitsneighborisalsogivenbyEq.( 7 ).MultipleTrustAuthorityScheme[ 23 71 ]:Inthisscheme,therearePsecuredomains.Ineachsecuredomain,thereexistsoneoff-linetrustauthority,whichcreatespublic-privatekeypairsforeachnodebelongingtoitsdomain.Eachnodebelongstokrandomlyselecteddomains.Therefore,twonodeshaveaprimarySAiftheybelongtothesamedomain,withprobabilitypf,whichisgiveninEq.( 7 ).NotethattheprocessesofneighborauthenticationandpairwisekeyestablishmentbasedonprimarySAsandthewaystosecuremorephysicallinkswithSLAhavebeenpresentedinauniedapproachinourpreviouswork(cf.[ 168 ,SectionII.B]),whichisomittedhereduetospaceconstraints.Here,wejustemphasizethatweassumeahomogeneoustrustmodel,i.e.,eachpairofnodeshasaprimarySAwiththesameprobabilitypf,andpf'sarepairwiseindependentforeachpairofnodes.Tosumup,wehavethefollowingobservationsfromtheschemesdiscussedabove.First,weneedkeeppfassmallaspossible.Alargerpfrequiresmorememoryspaceforstoringkeyingmaterialsineachnode,andwhenthatnodeiscompromised,therevealedkeyswillhavealargerimpactonnetworksecurity.Therefore,withthesame 204

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networkperformance,weareinterestedintheschemewiththeminimalpf.Second,alltheseschemesassumehomogeneousandindependenttrustrelationshipsamongnetworknodes,i.e.,everynodepairhasthesameprobabilitypfofsharingaprimarySAwhichhappensindependentlyofothernodepairs.Althoughthesetwopropertiesarenotnecessarilyvalidinallpracticalsituations(cf.[ 168 ,SectionII.A.2]),weadheretotheseassumptionsthroughoutthechapterfortheanalyticaltractability. 7.2.2OnSecureConnectivitySecureconnectivityherereferstotherequirementthatthereshouldexistasecurepathconnectingarbitrarynodepairs3,whichindicatestheavailabilityofsecurecommunications.Primaryresultsanalyzingsecureconnectivityhavebeenpresentedin[ 38 ]basedonanapproximationmodelforsensornetworks.Morepreciseanalysisaregivenin[ 36 70 ].Theseresultssufferfromtwomaindrawbacks.First,theyassumethatinordertoachievethesecureconnectivity,thenetworkshouldatleastbeconnectedwithprimarysecurelinks,whichisnotnecessarilythecase.Second,theyonlystudythecaseundercertainrequirementonpfforagivenrn=(p logn=n),thecommontransmissionrange.Inourpreviouswork[ 168 ],weovercometheselimitationsbygivingathoroughstudyonrn-pftradeoffswiththesecureconnectivityconstraintundertheassumptionsthatnnodesarerandomlydistributedinaunitareaandthatprimarySAsarepre-distributedasdescribedabove.Ourmainresultsareasfollows: ThenetworkissecurelyconnectedwithoutSLAorwithone-hopSLAwhenpfnr2n=(logn); 3Asecurepathconsistsofconsecutivesecurelinks.OfcoursethisrequirementisreasonableonlywhentheS-DpairisinthesametrustdomainandthereexistsatleastonephysicalpathconnectingtheS-Dpair.Therefore,itisnecessarytohavepf=(logn=n)andrn=(p logn=n)(cf.[ 168 ,SectionII.C]).Inwhatfollows,wealwaysassumethatitisthecase. 205

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Thenetworkissecurelyconnectedwithkn-hopSLAwhenpfnr2nc,wherec=(1)andkn=O(logn); Itisimpossibleforthenetworktobesecurelyconnectedwhenpfnr2n
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contributingtothistopic.Ourworkisdoneconcurrentlywithandindependentlyoftheworkin[ 14 ]anddifferentiatesitselffrom[ 14 ]asfollows.Firstofall,itisworthnotingthattherearesomefundamentaldifferencesbetweenmulti-channelWANETsandsecureWANETs.Iftwoneighboringnodesinamulti-channelWANETdonotshareacommonassignedchannel,thereexistsnophysicallinkbetweenthem.Incontrast,iftwoneighboringnodesinasecureWANETdonotshareacommonkey,itonlymeansthattheycannotestablishaprimarysecurelink.Thephysicallinkstillexistsandcanbesecuredandutilizediftheycanndaconnectedcommonfriendtohelpthem.Alsonotethat,iftwoconcurrenttransmissionpairsinamulti-channelWANETusedifferentchannels,theywillnotinterferewitheachother.Bycomparison,eventwoconcurrenttransmissionpairsinasecureWANETusedifferentkeystosecuretheirtransmissions,itisstillpossibleforthemtointerferewitheachother.Bytakingthesedifferencesintoconsideration,weshowthat,whenSLAisutilized,asecurethroughputof(1=p nlogn)isachievable,whichismuchhigherthantheresultin[ 14 ]forpf=(1=logn).Second,weadoptdifferentmodelsmoresuitableforsecureWANETs.Followingpreviousworks,e.g.,[ 49 60 100 101 140 154 ],theresultsin[ 14 ]and[ 13 ]areimplicitlybasedontheuidmodel,inwhichthepacketsareallowedtobearbitrarilysmallasn!1.Incontrast,wefollow[ 48 50 90 ]andexplicitlyassumetheconstant-packet-sizemodel,wherethepacketsizeremainsconstant,i.e.,doesnotscaledownwithn.Althoughtheanalysisoftheconstant-packet-sizemodelismuchharderthanthatoftheuidmodel[ 50 ],westillprefertheformersinceinrealitythepacketsizedoesnotchangewhenmorenodesareaddedtothenetwork.Furthermore,foraWANETwithsecurerequirements,eachpacketincludesamessageauthenticationcodeofatleastconstantsizeforcryptographicoperations.Thissecurityoverheadonthepacketlevelcanbeignoredasymptoticallyonlywiththeconstant-packet-sizemodel. 207

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Theadoptionoftheconstant-packet-sizemodelalsofacilitateouranalysisonpacketdelays[ 48 50 90 ].Finally,weutilizedifferenttechniquestoderivemoregeneralresults.Wedemonstratehowtotakeadvantageoftheconsiderablesimilaritybetweenourproblemandexistingworkonparallelcomputingandleveragetheresultsonfaultyarraystoobtainscalinglawsonsecurethroughputanddelay.Ourresultsactuallyapplytoallpossiblepf'swhenpf=(logn=n). 7.3SystemAssumptionsandMainResults 7.3.1RandomNetworkModelofWANETs 7.3.1.1NodedistributionWearemainlyinterestedinstaticWANETsornetworkswithslowmobility,inwhichtheround-triptime(RTT)ofapacketbetweenanyS-Dpairismuchsmallerthanthetimescaleofnetworktopologychanges.WedonotconsidertheWANETswithrapidtopologychangesbecauseinthiscasetheoverheadofmaintainingend-to-endpathswilldominatewirelesstransmissions,whileinthischapterwefocusontheoverheadintroducedbysecurityrequirementsanditsimpactondatatransmissions.Wemodelthenodepositionsasarandompointprocessasfollows.LetfX1,X2,gbeindependentanduniformlydistributedrandompointsonaboundedregionAintheplane.Givenapositiveintegern,thepointprocessfX1,X2,,Xngisreferredtoastheuniformn-pointprocessonAanddenotedbyXn.Givenapositivenumber=n jAj,letPo()beaPoissonrandomvariablewithparameter,independentoffX1,X2,g.ThenitcanbeshownthatthepointprocessfX1,X2,,XPo()gisaPoissonpointprocesswithmeanonA[ 130 ,Section1.7,p.18]andisdenotedbyP.Itisassumedthroughoutthechapterthatforanynor,therandompointprocessesXnandParecoupledinthismanner.VnisshorthandeitherforXnorP.RecallthatPischaracterizedbythefollowingspatialindependenceproperty:ifA1,A2,,AmarearbitrarilydisjointregionsofA,thenthenumbersofpointsinPonA1,A2,,Am 208

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aremutuallyindependentPoissonrandomvariableswithmeanjA1j,jA2j,,jAmj,respectively.Becauseofthisextremeindependenceproperty,itisofteneasiertoworkwithPratherthanXn.Therefore,weshalloftenstartbyprovinglimittheoremsaboutPas!1andthendeduceresultsforXnfromthese.Therationalebehindthisde-Poissonizationtechnique(cf.[ 130 ,Section2.5,p.37])isthat,giventhatthereareexactlykpointsofPinaregionA,thesekpointsareindependentlyanduniformlydistributedinA.Thus,XncanbewellapproximatedbyPasnortendstoinnity.NotethattheresultsobtainedinthischapterapplytobothXnandP(i.e.,Vn).WefurtherassumethatAisatorus4withaunitareaandtake=nasjAj=1,whichcorrespondstothedensenetworkmodel[ 48 60 ]becausetheareaisxedandthedensityofnodesincreaseswiththenetworksizen.Anotherpossiblemodelthatcanbeusedtostudytheasymptoticbehavioroflarge-scaleWANETsistokeepthenodedensityasaconstantandlettheareaofAincreaselinearlywithn,whichcorrespondstotheextendednetworkmodel[ 34 114 ].Inthischapterweconcentrateonthedensenetworkmodeljustforfaircomparisons,asmostknownresultsaboutWANETswithoutsecurerequirementsarebasedonthismodel[ 48 50 60 90 ].Wenotethat,however,ourresultscanalsobeappliedtotheextendednetworkmodelbyutilizingthescalingtechniqueintroducedin[ 114 ,Section2.2,p.28]. 7.3.1.2InterferencemodelsWeadoptthefollowingtwowidelyusedmodels[ 60 ]todescribethenecessaryandsufcientconditionforthesuccessfulreceptionofatransmissionoveronehop.Inwhatfollows,weassumethattimeisslottedforpacketizedtransmissionsandthatonlyO(1)packetscanbetransmittedpertimeslot,i.e.,ouranalysisisexplicitly 4Weassumethetorustoavoidbordereffects,whichotherwisecomplicatestheanalysis.Wenote,however,thattheresultsinthechapterholdforsquare,diskoranyothershapesofpracticalinterests. 209

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basedontheconstant-packet-sizemodel.AtransmittersendsdataataconstantrateofWpackets/timeslotforasuccessfultransmission,andzeroforanunsuccessfultransmission,whereW=O(1).ProtocolModel:Weassumethatallnodesuseacommonrangernfortheirtransmissions,andatransmissionfromnodeitonodejissuccessfulifandonlyifdijrnanddkj(1+)rnforanyothersimultaneoustransmitter,saynodek.Here,dijisthedistancebetweennodesiandj,andisapositiveconstantindependentofn.PhysicalModel:WeassumethatallnodesuseacommonpowerPnfortheirtransmissions,atransmissionfromnodeitonodejissuccessfulifandonlyifforaconcurrenttransmittersetS,wehavethesignaltointerferenceplusnoiseratio(SINR)atreceiverj,denotedasSINRij,satisfyingSINRij=PnGij N0+Pk2SnfigPnGkj.Here,istheSINRthreshold,N0representstheambientnoise,andGijdenotesthelinkgainonlinki!j.WeuseGij=d)]TJ /F19 7.97 Tf 6.59 0 Td[(ijforsimplicity,where>2isthepathlossexponent.Wemainlyfocusontheprotocolmodelinthischapterforacleanerpresentationofthekeyideas.WealsoshowthatthesameresultsonsecureWANETscanbeobtainedunderthephysicalmodelinAppendixD 7.3.1.3TrafcpatternSimilartopreviousworks[ 48 50 90 ],weconsidertheuniform-permutationtrafcpattern,i.e.,therearenows/sessionsandeachnodeisasourcenodeforonlyoneunicastsessionandadestinationnodeforanotherunicastsession.Supposethatthesourcenodei2f1,,nghasdataintendedfordestinationnoded(i),andthend(1),d(2),,d(n)isarandompermutationof1,2,,n,whered(i)6=iforalli. 210

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7.3.2NetworkPerformanceMetrics 7.3.2.1(Secure)throughputAper-owthroughputissaidtobefeasible/achievableifeverynodecansendatleastatarateofpackets/timeslottoitschosendestination.WedenotebyT(n),themaximumfeasiblethroughputasthethroughputcapacityforthenetwork.Whensecurityrequirementsareenforced,wedenesecurethroughputasthemaximumthroughputthatcanbesupportedonsecurepathsforallS-Dpairs.NotethatwhenSLAisutilized,thetrafcoverheadforconstructingsecurepathswillbeexcludedfromthetotaltrafconsecurepaths,sosecurethroughputisonlymeasuredasthedatarateachievedontheapplicationlayer. 7.3.2.2(Secure)delayThedelayofapacketisthetimeittakesthepackettoreachthedestinationafteritleavesthesource.Wedonottakethequeueingdelayatthesourceintoaccount,sinceourinterestisinthenetworkdelay.WeareinterestedintheexpectationoftheaveragepacketdelayoverallS-Dpairsandallrandomnetworkcongurations,whichisdenotedasD(n)throughoutthechapter.NotethatforsecureWANETs,thesecuredelayismeasuredonlyonsecurepaths.IfSLAisutilized,thetimerequiredtoconstructthesecurepathforthepacketgoingthroughthispathwillbecalculatedasapartofsecuredelayofthatpacket. 7.3.2.3ThepriceforsecurityThelossonthesecurethroughputortheincreaseonthesecuredelaycomparedtoWANETswithoutsecurerequirementswillbedenedasthepriceforsecurity. 7.3.3MainResultsofOurWorkThegoalofthischapteristostudytheimpactofrnandpfonthesecurethroughputanddelayofrandomnetworksdenedinSection 7.3.1 .Thefollowingresultsholdwith 211

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Figure7-1. Impactofsecurityrequirementsonthroughputscalinginrandomnetworks.Theshadedarearepresentsthroughputlossduetosecurerequirements.Thescalesoftheaxesareintermsoftheordersinn. highprobability(w.h.p.)5whenthenetworksizen!1.Hereweonlyconsiderthesituationwhenpf=(logn=n)andrn=(p logn=n)(cf.Footnote 3 ). Theorem7.1. Whenpf=(logn=n),thesecurethroughputwithoutSLAisT(n)=q pf nlognpackets/timeslot(segmentA-BinFigure 7-1 ),andthesecuredelayisD(n)=q npf logn. Theorem7.2. (i)Whenpf=(logn=n)andalsopf=O(1=logn),thesecurethroughputwithSLAisT(n)=(p pf=n)packets/timeslot(segmentC-DinFigure 7-1 ),andthecorrespondingdelayisD(n)=(p n=pf);(ii)Whenpf=(1=logn),thesecurethroughputwithSLAisT(n)=(1=p nlogn)packets/timeslot(segmentD-EinFigure 7-1 ),andthecorrespondingdelayisD(n)=(p n=logn).ComparingTheorem 7.1 withTheorem 7.2 ,wecanconcludethat,SLAisnecessarybecauseingeneralitcanincreasetheachievablethroughputasafactorof(p logn).However,itdoesnotmeanthatweshouldtrytosecureallphysicallinks.RememberthatSLAalsoincursextracommunicationoverhead,andtheschemewedesigntoachievethethroughputinTheorem 7.2 showsthatweneedcarefullychoose 5Herew.h.p.referstoaprobabilityatleast1)]TJ /F7 11.955 Tf 14.12 0 Td[((n),forafunction(n)goingto0withn!1. 212

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linkstobesecuredwiththehelpoffriendsinordertoguaranteethatthebenetsfromSLAalwaysexceeditscosts(cf.Section 7.5 ).Inordertocalculatethepriceofsecurity,werecalltheresultsonnetworkperformanceofWANETswithoutsecurityrequirements[ 48 50 90 ],whichcanbesummarizedasthefollowingTheorem. Theorem7.3. ThethroughputcapacityofWANETswithoutsecurityrequirementsisT(n)=(1=p nlogn)packets/timeslot(dashedlinesinFigure 7-1 ),andthecorrespondingdelayisD(n)=(p n=logn).6FromTheorem 7.3 ,wecanndthatthepriceofsecuritymainlyexhibitsinthelossofthroughput.Figure 7-1 givesanillustrationofthecomparisononthroughputcapacitywithorwithoutsecurityrequirements.Itisworthnotingthatwhenpfis(1=logn),securitycomeswithnopriceinasymptoticsense,i.e.,securenetworkperformanceremainsonthesameordercomparedtothenetworkswithoutsecurityrequirements(cf.Theorem 7.2 (ii)).Webelievethatthisresultisquiteimportantbecauseitprovidesvaluableinsightonthedesirableoperatingpointsthatbalancesecurityandefciencyconcerns.Weneedtominimizepfinordertoreducethememorysizeforkeyingmaterialsandmitigatetheimpactofnodesbeingcompromised.Ourresultimpliesthatevenwhenpfgoestozeroasthenetworksizebecomesarbitrarilylarge,aslongaspf=(1=logn),itisstillpossibletosecurealarge-scaleWANETwithnegligibleoverhead.Ourresultsalsoshowthatsecurityrequirementsingeneralwillnotincrease 6Thethroughputcapacityof(1=p nlogn)wasrstlyprovedbyGuptaandKumarin[ 60 ],buttheiranalysisisbasedontheuidmodel.ThesameresultwasobtainedbyKulkarniandViswannath[ 90 ]throughtheconstantpacketsizemodel.ElGamaletal.[ 48 50 ]furtherimprovedthisresultbygivingboundsonD(n).NotethatrecentlyFranceschettietal.[ 46 ]showedthatthe(1=p n)throughputcapacityisachievableifwerelaxtheassumptionthatallnodesusethesamern.Herewestilluse(1=p nlogn)boundonthroughputbecauseourtrustmodelisahomogeneousone,andforafaircomparison,wealsoassumetherandomnetworkmodelishomogeneous,i.e.,allnodeshavethesamern. 213

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Figure7-2. ThesecurecommunicationschemewithoutSLA. thee2edelay.Thiscanbeintuitivelyexplainedasfollows:inordertokeepthesecureconnectivity,whichistheprimaryrequirementforsecureservices,thenegativeeffectofasmallerpfshouldbecompensatedbyalargerrn,whichwilleffectivelydecreasethenumberofhopsapacketneedtravelandthusthee2edelay. 7.4NetworkPerformancewithoutSLAWenowpresentaparameterizedsecurecommunicationschemewithoutSLAandanalyzeitsperformance.OurtheoreticalresultsinTheorem 7.4 conrmthattheboundsgiveninTheorem 7.1 areachievableandtight. 7.4.1SchemeDescriptionScheme1:thesecurecommunicationschemewithoutSLA (1) TorusPartition:Dividetheunittorusintoasetofregularcells,eachofsidelengthcn=q c1logn npf,wherec1isaconstant.SeeFigure 7-2 (a)foranillustration. (2) SettingTransmissionRange:Setrn=p 5cn,whichguaranteesthateachnodecandirectlycommunicatewithanynodeinthesamecellorintheimmediateverticalandhorizontalneighboringcells. 214

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(3) Routing:Packetsaredeliveredfromthesourcetothedestinationintwophases.First,theyareforwardedalongthecellsintherowthatcontainsthesourcecelluntiltheyreachthecolumnthatcontainsthedestinationcell.Inthesecondphase,packetsareforwardedalongthecellsinthesamecolumntotheirdestination.TheL-shapedcurveconnectingthesourceanddestinationasdescribedaboveiscalledS-Droutes(shadedareainFigure 7-2 (a)). (4) CellScheduling:Acellulartime-divisionmulti-access(TDMA)transmissionschemeisused,inwhicheachcellbecomesactive,i.e.,itsnodescantransmitsuccessfullytonodesinthesamecellorinneighboringcells,atregularlyscheduledtimeslots(cf.Proposition 7.1 ). (5) PacketTransmissionScheduling:Eachpacketwillhaveatimestamptbdenotingthetimeslotthepacketwastransmittedbythesource.Whenacellbecomesactive,itwillselectthepacketwiththesmallesttbinthecelltotransmit.Ifthereareties,choosethepacketfromtheS-Dpairiwhichmaximizes(tb+i)modn.Notethatonlyonepacketistransmittedpertimeslotpercell.7Ourpackettransmissionschedulingschemewilltreatthepacketsfromdifferentsessionsequallyandprefertheoldestpacketineachsession. (6) SecureInter/Intra-CellTransmission:Allthepacketswillbetransmittedonprimarysecurelinks.Whenanodeneedtransmitapackettoitsneighboringcell,italwaystransmitsthepackettooneofitsfriendsinthatcell(secureinter-celltransmission);otherwise,itwilldropthepacket.Whenanodeneedtransmitapackettothenodeinthesamecellandtheyarenotfriends,thenodewillndoneoftheircommonfriendsinthesamecellastherelaynode(secureintra-celltransmission).Ifitcannotndone,itwilldropthepacket.Herewegivesomeprimaryresultsontheschemedescribedabove.WerstrecallthefollowingresultonthepropertyofcellschedulinginStep(4),whichiswidelyknownnow[ 90 ]. Proposition7.1. Undertheprotocolmodel,thereexistsaninterference-freeschedulesuchthateachcellbecomesactiveregularlyonceinK2timeslotswithoutinterferingwithanyothersimultaneouslytransmittingcell.HereKdependsonlyonandisindependentofn.SeeFigure C-1 foranexampleofthisinterference-freeschedule. 7EachS-Dpairisidentiedbythesourcenode'sID. 215

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Next,weshowthattheprobabilitythatthescheduledpacketisdroppedinStep(6)inScheme1approachestozeroasn!1.Thisclaimistrueduetothefollowinglemma. Lemma10. InScheme1,wecanalwaysndaconstantc1suchthat(i)eachcellcontains(logn=pf)nodesw.h.p.;(ii)givenanarbitrarynodei,eachcellcontains(logn)friendsofiw.h.p.;(iii)giventwoarbitrarynodesiandjinthesamecell,eithertheyarefriendsortheyhaveatleastonecommonfriendinthatcellw.h.p. Proof. SeeAppendixC Lemma 10 showsthat,foranysourcenodeS(Figure 7-2 (a)),itcanalwaysndafriendineachneighboringcellsw.h.p.Ifmultiplefriendsareavailableinacell,Srandomlychoosesoneanddenesthisfriendasitssecurerelayinthiscell.WiththeroutingruleinStep(3),thisfriend-ndingprocedure(i.e.,eachsecurerelaynditsownfriendsinthefollowingneighboringcells)iscontinueduntilthereisasecurebackbone(regularsolidlinesinFigure 7-2 (a))spanningallcells.BasedonLemma 10 (ii),everynodecanconstructitsownsecurebackbone8asdescribedabove.Therefore,eachpacketcanfollowthissecurebackboneuntilitreachesthesecurebackbonenodekinthesamecellasthedestinationnodeDthroughsecureinter-celltransmissions.IfkisafriendofD,itcandirectlytransmitthepackettoD;otherwise,itneedndacommonfriendjtorelaythepacket,whichisguaranteedtohappenw.h.p.accordingtoLemma 10 (iii).Therefore,thereareatmosttwosecureintra-celltransmissions(boldsolidlinesinFigure 7-2 (a))foreachpacketw.h.p. 8Notethatthesourcenodedoesnotneedtoconstructthissecurebackbonebeforehand.Itwillemergegraduallywiththedataowandextendtoanewcellwhenthesourcenode'spacketgoesthroughthatcell.HerewejustshowthatthereexistssuchasecurebackboneforeachnodetofacilitatetheanalysisinSection 7.4.2 216

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7.4.2PerformanceAnalysisofScheme1NotethatScheme1onlyoperatesonprimarysecurelinks.Theselinkswillbeestablishedafterdirectneighborauthenticationoperations,whichonlyneedlocalbroadcasts.Obviouslytheoverheadandthedelayincurredhereisnegligiblecomparedtomulti-hopdatacommunications(cf.[ 168 ,SectionII.B]).Therefore,inwhatfollows,weonlyconcentrateonthethroughputanddelayinthedatadeliveryphase.OuranalysisonScheme1mainlyreliesonsomewell-knownresultsabout2-dimensional(2-d)arrays[ 80 94 ],whichhavebeenextensivelystudiedintheparallelanddistributedcomputingresearchcommunity.Therefore,werstreviewsomerelateddenitionsandresults.A2-dllarrayconsistsofL=l2processorsorprocessingunits(PUs)arrangedina2-dllgrid.EachPUisconnectedtoitsfourneighborsviapoint-to-pointwiredcommunicationlinks.IntheMIMD(MultipleInstructionMultipleData)mode,thePUsperformroutinginaseriesofsynchronoustimeslots.Duringeachtimeslot,aPUmaysendonepackettoitsneighborsalongeachofthe(uptofour)linksincidentonit.APUmayalsoreceiveonepacketalongeachofitsincidentlinksduringatimeslot.ThePUscanbeindicatedbytheircoordinateswithinthearray;thePUatposition(i,j),0i,jl,isdenotedPi,j.Hereposition(0,0)liesintheupper-leftcorner(refertoFigure 7-2 (b)foranillustration).Atorusisanarraywithso-calledwrap-aroundlinks,whichconnectPi,0withPi,l)]TJ /F9 7.97 Tf 6.58 0 Td[(1andP0,jwithPl)]TJ /F9 7.97 Tf 6.59 0 Td[(1,j.Throughoutthischapter,allresultsaboutanarraycanbeextendedtothecorrespondingtorus,sowedonotdistinguishtorifromarrayshereafterandsimplycallthemarrays9forsimplicity.Anh-hroutingproblemon2-darraysreferstothescenariosthateachPUisthesourceanddestinationofexactlyhpackets. 9Alsonoticethat,throughoutthischapterweonlyconsider2-darrayswithintheMIMDmode. 217

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Lemma11. ([ 94 ]and[ 80 ])h-hroutingonllarrayscanbeperformeddeterministicallyinhl=2+O(h5=6l2=3)timeslots,withaveragepacketdelay(l).WenowpointoutthecorrespondencebetweenScheme1andtheoptimalcommunicationschemeforthe2-darray.Let l=1 cn=r npf logn.(7)CellCi,jinFigure 7-2 (a)correspondstoPUPi,jinFigure 7-2 (b)for0i,jl.Withoutlossofgenerality,weassumethateachsourcenodehasonlyonepacketinourWANETmodel,sothereare(logn=pf)packetsgeneratedineachcellbasedonLemma 10 (i).Byletting h=(logn=pf),(7)wehaveformedacorrespondenceinthetrafcpatternbetweenourWANETmodelandthearray,i.e.,weassociatethehpacketsgeneratedinaPUwiththepacketsofthenodescontainedinthecorrespondingcell.RoutingandschedulingalgorithmsusedbythearraytoachievetheperformancegiveninLemma 11 arethesameastheschemeswedescribedinStep(3)and(5),respectively.Infact,Scheme1simulatestheoptimalcommunicationschemeforthearraybyrequiringtheschedulednodeincellCi,jtoperformthecommunicationoperationperformedbyPUPi,jofthearray.Next,wediscussthedifferencebetweenourWANETmodelandthearray.NotethateachPUcantransmitandreceiveupto4packetsineachtimeslot,whileinourcellschedulingschemeinStep(4),eachcellwillbescheduledtobeactiveonceinK2timeslots(cf.Proposition 7.1 ).Therefore,comparedtothearray,theschemeperformedinWANETwillhaveaslowdownbynomorethanafactorof4K2.Therefore,wemakeacorrespondencebetweentheperformanceofsecureinter-cellcommunicationswiththatofcommunicationsbetweenneighboringPUs.BasedonLemma 11 ,wecanconcludethatthetotalnumberoftimeslotsneededtodelivern 218

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packets(oneforeachsourcenode)totheirdestinationnodes'cellsequalto 4K2hl 2K=(1)=(hl)( 7\0003 ),( 7\0004 )= s nlogn pf!.(7)WehavealreadyshowninSection 7.4.1 thateachpacketonlyneedsatmost2secureintra-celltransmissions,correspondingto2K2timeslots.Therefore,thetotalnumberoftimeslotsneededtodelivernpacketstotheirdestinationnodes,denotedas'(n),canstillbeexpressedinEq.( 7 ).Sinceonlyonepackethasbeendeliveredforeachnode,weobtainper-nodethroughputas=1='(n)packets/timeslot.Sinceweassumetheconstant-packet-sizemodelandthatonepacketcanbetransmittedineachtimeslot,wehaveT(n)=1='(n)packets/timeslot.FromLemma 11 andEq.( 7 ),wecandirectlyobtainD(n)=(p npf=logn)timeslots.Basedontheaboveanalysis,wehaveinfactgivenaconstructivelowerboundonT(n)andupperboundonD(n)withoutSLAasfollows. Theorem7.4. Whenpf=(logn=n),thesecurethroughputwithoutSLAisT(n)=q pf nlognpackets/timeslotandthecorrespondingdelayisD(n)=Oq npf logntimeslots.Inparticular,whenpf=1orwithoutanysecurityrequirement,wecanobtainGuptaandKumar'sresult[ 60 ].ElGamaletal.[ 48 50 ]reprovetheirresultundertheconstant-packet-sizemodelwithcomplicatedanalysisonadiscrete-timequeuingnetwork.Here,wefollowKulkarniandViswannath'smethodology[ 90 ]toavoidthesecomplicatedqueueinganalysisbyexploitingthesimilaritybetweenthecell-basednetworkmodelandthearray.Therefore,ourproofisdesirableinitssimplicity.Moreover,itprovidessomenecessarybackgroundforunderstandingourmorecomplicatedschemedesignedwithSLA. 219

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7.5NetworkPerformancewithSLAInthissection,weanalyzeachievablesecurenetworkperformancewhenSLAisallowed.WerstpresentthefollowingschemestoachievetheperformanceboundsinTheorem 7.2 7.5.1SchemeDescriptionAsapreludetodescribingthescheme,wereviewtheSLAoperationsdenedinourpreviouswork[ 168 ].Whenpfnr2nc2forsomeconstantc2,thenetworkconsistingofnodesandprimarysecurelinks(modeledasprimarysecuregraph)isinthepercolatedphase,i.e.,mostnodesareconnectedbyasecurebackbone(alsocalledthegiantclusterinpercolationtheory)withprimarysecurelinks.Therearestillpinnodesdisconnectedfromthegiantcluster,where0
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Figure7-3. Multi-hopSLAoperations.HeresolidlinesanddashedlinesrepresentprimarysecurelinksandprimarySAs,respectively.Solidandopenpointsrepresentnodesonthegiantcluster(securebackbone)ornot,respectively.Nodeiisisolatedandneedstobeconnectedwiththesecurebackbonewiththehelpofitsfriend,e.g.,nodek. (4) Nodesiandjmutuallyauthenticateeachotherandsecurethephysicallinki$j.OneimportantpropertyofScheme2weobtainedin[ 168 ]isthatnodekisO(logn)hopsawayfromnodej.Orputitinanotherway,inordertondafriendofnodeiinthesecurebackbone,weneedvisitO(logn)nodesw.h.p.AlsonotethatoneprerequisiteofScheme2isthateverynodeshouldknowwhetheritisanisolatednodeoranodeinthesecurebackbone.Thisnecessitatesasecurenetworkpartitiondetectionalgorithmperformedineachnodetodecideitsroleintheprimarysecuregraph.Ourpreviousresearch[ 168 ]showsthatinthepercolatedphase,isolatednodesonlyformclusterswithsizeO(1)evenwhenn!1.Therefore,eachnodecansendaprobemessage,whichwillbeforwardedonlythroughprimarysecurelinks.IftheprobemessagecanonlygothroughO(1)hops,w.h.p.thenodeisisolated.Theassociatedoverheadforthesecurenetworkpartitiondetectionisonthesameorderofdirectneighborauthentication,astheybothrequirecommunicationswithinO(1)hops,whichcanbeignoredascomparedtothenetwork-widemulti-hopcommunications.Next,wepresentaprimarysecurecommunicationschemeforS-Dpairsonthesecurebackbone,andourtaskistodeliverapacketfromthesourcenodetothesquareletinwhichthedestinationnodedwells.Scheme3`:securecommunicationsonthesecurebackbone 221

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Figure7-4. Securecommunicationscheme3`withSLA.In(a),solidlinesanddashedlinesrepresentprimarysecurelinksandprimarySAs,respectively.TheshadedarearepresentsthesquareletsthatcanbecoveredbythesecurebackboneofsourcenodeS.In(b),solidpointsandopenpointsrepresentPUsonandoffthegiantcluster,respectively, (1) TorusPartition:Dividetheunittorusintoasetofregularsquarelets,eachofsidelength sn=8<:q c3logn nifpf=1 logn,q c3 npfotherwise, (7) wherec3isaconstant.NotethatacellusedinScheme1ismuchlargerthanasquareletingeneral.Infactacellcontains(logn)squareletswhenpf=o(1=logn)(refertoFigure 7-4 (a)foranillustration). (2) SettingTransmissionRange:Setrn=p 5sn,whichguaranteesthatnodesinneighboringsquareletscancommunicatedirectly.Alsonoticethatthisrnguaranteesthattheprimarysecuregraphisinthepercolatedphasew.h.p.(thisisadirectconsequenceofTheorem2inourpreviouswork[ 168 ]). (3) SquareletandPacketTransmissionScheduling:ThesquareletschedulingandthepackettransmissionschedulingineachactivesquareletarethesameasthecellschedulinginStep(4)ofScheme1,andthepacketschedulingineachactivecelldescribedinStep(5)ofScheme1,respectively. 222

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Figure7-5. Routingschemeonthepercolatedgrid.HeresolidlinesrepresentsecurelinksbetweenneighboringPUs/squarelets.SolidandopenpointsrepresentPUsonandoffthegiantcluster(securebackbone),respectively.TheshadedarearepresenttheniteclustersofPUsdisconnectedfromthesecurebackbone,anddashedlinesrepresentbordersoftheseclusters. (4) SecureInter-SquareletTransmission:Allthepacketswillbetransmittedonprimarysecurelinkscrossingneighboringsquarelets.Inotherwords,whenanodeneedstotransmitapackettoitsneighboringsquarelet,italwaystransmitsthepackettooneofitsfriendsinthatsquarelet.AswhatwehavedoneinScheme1,wecanestablishacorrespondencebetweenoursquareletsystemandthellarraybysettingl=d1=sne.SeeFigure 7-4 foranillustration.Here,twoneighboringPUswillhavealinkinthearray,ifthepacketholderinthecorrespondingsquareletcanndafriendinanothersquarelet.Therefore,thearrayweobtainedinFigure 7-4 (b)isafaultyarray(whichwillbedenedmorepreciselysoon)withlinkfailures,whereafailureindicatesthereisnofriendinaneighboringsquarelet. (5) Routing:Sincehereweonlyconsidertheinter-squareletcommunications,routingbetweensquareletsisequivalenttothatoperatingonPUs.Therefore,weusethefaultyarrayasanexampleforacleanerpresentation.Notethatthecorrespondingarrayisalsopercolated,andwecanguaranteethatthereexistsapathconnectingtheS-DPUsifthecorrespondingS-Dnodepairsareonthesecurebackbone.SeeFigure 7-5 foranillustration.WerstxoneshortestpathoflengthkconnectingtheS-DPUsinthearraywithoutfaultylinks,whichconsistsofsquarenodesinFigure 7-5 .Ourroutingalgorithmattemptstofollowthisshortestpathuntilitencountersafailurelink,e.g.,atnodei.Atthispoint,wesimplycircumnavigatestheclusterofisolatednodes(theshadedarea)thatblocksthepath,untileitherthedestinationPUisreached,orthealgorithmisbackontotheoriginalshortestpathtoit(e.g.,reachnodem).Sincetheaveragesizeoftheclusterofisolatednodesisaconstantw.h.p.,thepathlengthofroute(boldlinesinFigure 7-5 )foundbyourschemeisO(k)onaverage[ 112 ]. 223

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WenextsummarizesomebasicresultsabouttoruspartitioninScheme3`inthefollowinglemma. Lemma12. InScheme3`,wecanalwaysndaconstantc3suchthat(i)eachsquareletcontains)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(ns2nor(logn)nodesw.h.p.;(ii)eachsquareletw.h.p.con-tains)]TJ /F6 11.955 Tf 5.48 -9.69 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(pi)ns2nand)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(pins2nnodesonandoffthesecurebackbone,respectively,where0
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(3) Isolatednodestransmitthepacketsgeneratedbythemselvestotheirdeputies,respectively. Phase3-secure-backbonecommunication:AfterPhase2,allthepacketsgeneratedbysourcesareredistributedonsecure-backbonenodesonly,andthenwecanutilizeScheme3`todeliverthesepacketsfromthesourcenodesordeputiestotheircorrespondingdestinationsquarelets(greenlinesfromStoiinFigure 7-4 (a)).Moreprecisely,ifthedestinationnodeisalsoonthesecurebackbone,wedenethesquareletinwhichitdwellsasthedestinationsquarelet.Otherwise,wedeneoneoftheclosestsquareletstothedestinationnode,whichisalsocoveredbythesecurebackbone,asthedestinationsquarelet.Notethatthedestinationsquareletisalwayscoveredbythetransmissionrangeofthedestinationnode,whichisguaranteedbyourtoruspartitioninScheme3`. Phase4-last-hopdelivery:SeeFigure 7-4 (a)foranexample.IfnodeiisafriendofthedestinationnodeD,thenicandirectlytransmitthepackettoD.Otherwise,weneedtosecurelinki!D.ThiscanbedonebyutilizingScheme2again:wendafriendofD,saynodek,andthensecurethelinki!Dwiththehelpofk.AsdescribedinStep(2)ofPhase2,weneeduseScheme3`twicetofulllthisoperation. 7.5.2PerformanceAnalysisofScheme3WerstanalyzetheperformanceofScheme3`.Ouranalysismainlyreliesonthefollowingresultsonfaultyarrays.Aq-faultyarrayreferstothearrayinwhicheachlinkmayfailindependentlywithsomeprobabilityboundedabovebyaxedvalueq. Lemma13. Thereexistsaschemeforaq-faultyllarraytosolvethe1-1routingproblemin(l)timeslotswithprobability1)]TJ /F6 11.955 Tf 12.66 0 Td[(1=lwhenqissmallenough.Notethatforfaultyarrays,wearerequiredtoroutepacketsonlivelinks,andweonlyneedroutepacketsamongallPUsconnectedbylivepaths.Remark:MathiesprovesLemma 13 in[ 112 ]forq<0.5.ItistrivialtoextendLemma 13 totheh-hroutingproblem:inthesamewaywecanperformtheh-hroutingonaq-faultyllarraywithin(hl)timeslotswiththeaveragepacketdelayof(l)w.h.p.ComparedtoLemma 11 ,theresulthereshowsthatwhenq<0.5,therunningtimeforaq-faultyarrayisalmostthesameasiftherewerenofaultsinthearraylinks(uptoconstantfactors),ifaroutingschemesimilartotheonewedescribedinStep(5)inScheme3`isadopted.ItistrivialtondthecorrespondencebetweenourScheme3` 225

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andthe(1)]TJ /F3 11.955 Tf 12.34 0 Td[(pl)-faultyarray.Therefore,theaboveresultscanbeleveragedtoanalyzeScheme3`whenpl>0.5,whichcanbeeasilyachievedbytuningtheparameterc2mentionedinSection 7.5.1 .FollowingthesameargumentgiveninSection 7.4.2 ,weobtainthefollowingresult. Corollary7.1. WheneachnodeonthesecurebackbonehasatmostO(1)packets,Scheme3`candeliverallthesepacketswithin(nsn)timeslotswiththeaveragepacketdelayof(1=sn)w.h.p. Proof. ThiscanbedirectlyobtainedfromLemmas 12 and 13 WenowanalyzetheperformanceofScheme3.NotethatPhase1andSteps(1)and(3)inPhase2onlyneedlocalbroadcasts,whichwillbedominatedbyotherphasesinvolvingScheme3`.Wethusignoretheminourasymptoticanalysis.Step(1)inPhase2guaranteesthateverysecure-backbonenodewillactasthedeputyfor(jNIj=jNSj)isolatednodes.FromLemma 12 (ii),weknowthatitisequalto(1).Therefore,everysecure-backbonenodeonlyneedhandle(1)SEC-REQorSEC-APVmessages.ThentheperformanceofScheme3`usedinStep(2)ofPhase2canbeboundedasinCorollary 7.1 .Forthesamereason,thenetworkperformanceinPhase4isalsoboundedasinCorollary 7.1 .FromSteps(1)and(3)ofPhase2,wecanguaranteethateachsecure-backbonenodeonlyholds(1)packetsatthebeginningofPhase3,assumingthateachsourcenodeonlygeneratesonepacket.Therefore,wecanapplyCorollary 7.1 againtoPhase3.Tosumup,theperformanceofScheme3isonthesameorderofthatofScheme3`,whichischaracterizedbyCorollary 7.1 .BysubstitutingEq.( 7 )intoCorollary 7.1 andfollowingtheargumentgiveninSection 7.4.2 ,wecanobtaintheboundsonthesecurethroughputanddelay.Basedontheaboveanalysis,wehaveinfactobtainedaconstructivelowerboundonT(n)andupperboundonD(n)withSLAasfollows. Theorem7.5. (i)Whenpf=(logn=n)andalsopf=O(1=logn),thesecurethroughputwithSLAisT(n)=(p pf=n)packets/timeslotandthecorresponding 226

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delayisD(n)=O(p n=pf)timeslots;(ii)Whenpf=(1=logn),thesecurethroughputwithSLAisT(n)=(1=p nlogn)packets/timeslotandthecorrespondingdelayisD(n)=O(p n=logn)timeslots. 7.6OptimalityofOurSchemesInthissection,wepresentupperboundsonthesecurethroughputwithorwithoutSLA.Thecorrespondinglowerboundsonthee2edelaywillalsobeobtained.SincetheupperboundsderivedherematchtheconstructivelowerboundsobtainedinSection 7.4 and 7.5 ,wecompletetheproofoftheTheorems 7.1 and 7.2 ofthischapterundertheprotocolmodel.TheresultsinthissectionalsoshowthattheschemeswedesignedinSection 7.4 and 7.5 areoptimalatleastintheordersense.NotethatwedefertheproofsofTheorems 7.1 and 7.2 underthephysicalmodeltoAppendixD 7.6.1UpperBoundsonSecureThroughputsThesecurethroughputofrandomnetworksdenedinSection 7.3.1 islimitedbythefollowingthreeconstraints.Themaximumfeasiblethroughputsatisfyingalltheseconstraintsisanupperboundonthesecurethroughput.Whiletheremaybeotherconstraintsundersecurethroughputaswell,theconstraintsweconsiderherearesufcienttoprovidetightbounds,astheupperboundsobtainedherematchtheconstructivelowerboundsprovidedinSection 7.4 and 7.5 .Physical-ConnectivityConstraint:Werstneedtomakesurethatthenetworkisphysicallyconnected,whichconstrainsrnasrn=(p logn=n)[ 59 130 ].Secure-ConnectivityConstraint:ThethroughputofsecureWANETsisconstrainedbytheneedtoensurethatthenetworkissecurelyconnected,sothateveryS-Dpaircancommunicatethroughatleastonesecurepath.Ourpreviouswork[ 168 ]quantiesthisconstraintasfollows(cf.Section 7.2.2 ): rn=q logn npfwithoutSLA; rn=(1=p npf)withSLA. 227

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InterferenceConstraint:Thesecurethroughputisalsoconstrainedbyinterference.Sincethewirelesschannelisasharedmedium,undertheprotocolmodel,twonodessimultaneouslyreceivingapacketfromdifferenttransmittersmustbeseparatedbyenoughdistance.ThisimpliesaconstraintonthemaximumnumberofsimultaneoustransmissionsintorusA.Wecharacterizethisconstraintwiththefollowinglemma. Lemma14. TheinterferenceconstraintrequiresthatT(n)c3 nrn,wherec3isaconstant. Proof. Werstconsiderthecasewhenpf=1.LetLbetheexpecteddistancebetweenS-Dpairswithintheunit-areatorus,andthenL=(1)w.h.p.(cf.[ 90 ,Claim3.1(3)]).Thusonaverageeachpacketneedtraverseatleast)]TJ /F9 7.97 Tf 7.88 -3.21 Td[(L rnhopstoreachthedestination.SinceeachnodegeneratespacketsatrateT(n),thismeansthatthepacketspertimeslotbeingtransmittedbythewholenetworkisatleastnT(n)L rn.Undertheprotocolmodel,eachtransmissionconsumesarea,i.e.,disksofradius 2rnaroundeverytransmittershouldbedisjoint[ 60 ].SincetheareaconsumedisboundedabovebythetotalareajAj=1,themaximumnumberoffeasiblesimultaneoustransmissionsisnomorethan4 2r2n.Hencewehavetheconstraint,nT(n)L rnW4 2r2n)T(n)c1 nrn.Thethroughputofnetworkwhenpf=1isatleastaslargeasthethroughputofthenetworkwhenpf<1(thisistriviallytrue,bynotusingunsecuredphysicallinks),soc1 nrnisalsoanupperboundforT(n)whenpf1. Bycombiningtheaboveconstraints,weobtainthefollowingtheoremontheupperboundsonthesecurethroughput: Theorem7.6. (i)Whenpf=(logn=n),thesecurethroughputwithoutSLAisT(n)=Oq pf nlognpackets/timeslot;(ii)Whenpf=(logn=n)andalsopf=O(1=logn),thesecurethroughputwithSLAisT(n)=O(p pf=n)packets/timeslot;(iii)Whenpf=O(1)andalsopf=(1=logn),thesecurethroughputwithSLAisT(n)=O(1=p nlogn)packets/timeslot. 228

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7.6.2LowerBoundsonSecureDelaysLowerboundsonsecuredelayscanbeanalyzedinasimilarfashion.Theonlythingweneedtodoistoreplacetheinterferenceconstraintwiththefollowingpath-lengthconstraint.Path-LengthConstraint:Sinceonlyasinglepacketcanbetransmittedpercellpertimeslot,thee2edelayislowerboundedbythenumberofhopsonthepath.Let LbetheexpecteddistancebetweenS-Dpairs.WethenhaveD(n) L rn.Ifwerequirethatthepacketisalwaystransmittedthroughthesecurepath,D(n)isevenlarger,therefore,D(n)=(1=rn).Bycombiningtheaboveconstraintwithphysicalandsecureconnectivityconstraints,weobtainthefollowingtheoremforthelowerboundsonthesecuredelay: Theorem7.7. (i)Whenpf=(logn=n),thesecuredelaywithoutSLAisD(n)=q npf logntimeslots;(ii)Whenpf=(logn=n)andalsopf=O(1=logn),thesecuredelaywithSLAisD(n)=(p n=pf)timeslots;(iii)Whenpf=O(1)andalsopf=(1=logn),thesecuredelaywithSLAisD(n)=(p n=logn)timeslots. 7.7ChapterSummaryInthischapter,basedonageneralrandomnetworkmodel,theasymptoticbehaviorsofsecurethroughputanddelaywiththecommontransmissionrangernandtheprobabilitypfofneighboringnodeshavingaprimarysecurityassociation,arequantiedwhenthenetworksizenissufcientlylarge.Thecostsandbenetsofsecurelinkaugmentationoperationsonthesecurenetworkperformancearealsoanalyzed. 229

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CHAPTER8CONCLUSIONANDFUTUREDIRECTIONS 8.1DissertationSummaryInthisdissertation,wehavestudiedseveralchallengingandfundamentalissuesrelatedtothenetworkperformanceandsecurityofwirelessadhocnetworks.Themaincontributionsofthisdissertationcanbesummarizedasfollows.Afterawirelesssensornetwork(WSN)isdeployed,sensornodesareusuallyleftunattendedforalongperiodoftime.ThereisaninevitabledevolutionoftheconnectedcoverageoftheWSNduetobatteryexhaustionofsensornodes,intendedphysicaldestructionattacksonsensornodes,unpredictablenodemovementbyphysicalmeanslikewind,andsoon.Itis,therefore,criticalthatthebasestation(BS)learninrealtimehowwelltheWSNperformsthegivensensingtask(i.e.,whatisthecurrentconnectedcoverage)underadynamicallychangingnetworktopology.Inthisdissertation,weproposeacoverageinferenceprotocol(CIP)whichcanprovidetheBSanaccurateandin-timemeasurementofthecurrentconnectedcoverageinanenergy-efcientway.Especially,weshowthattheschemecalledBONDwhichourCIPrequirestobeimplementedoneachsensornodeenableseachnodetolocallyself-detectwhetheritisaboundarynodewiththeminimalcommunicationandcomputationaloverhead.TheBONDcanalsobeexploitedtoseamlesslyintegratemultiplefunctionalitieswithlowoverhead.Moreover,wedeviseextensionstoCIPwhichcantoleratelocationerrorsandactivelypredictthechangeoftheconnectedcoveragebasedonresidualenergyofsensornodes.Networkheterogeneityisacertaintyfortoday'swirelessnetworks.Therearetwokindsofheterogeneity:rst,thedistributionofwirelessusers/devicesinthephysicalspaceisnon-homogeneous;second,wirelessdevicesarelikelytohavewidelyvaryingradioranges(e.g.,cellular/WiMax,WiFi,Zigbee).Foraheterogeneousadhocnetwork,whereshort-rangewirelesslinksandlong-rangewirelesslinks(shortcuts)coexist, 230

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howtodesignefcientdecentralizedroutingprotocolswithlocalinformationisanopenproblemintheliterature.Inthisdissertation,weshowthatthelong-rangelinksarenotnecessarilyhelpfulfordecentralizedrouting.Sometimes,shortpathswithlong-rangelinksexistinthenetwork,however,thedecentralizedroutingschemeswithonlylocalinformationcannotndthem.Weprovethatthenecessaryandsufcientconditionforlocalizedrouting(e.g.,greedygeographicrouting)tobeefcientisthattheprobabilityofalong-rangelinkbeingpresentfromnodeutovshouldbeinverselyproportionaltothenumberofnodeswhichareclosertouthanvis.Ourresultshowsthatitisthedistributionpatterninsteadoftotalnumberofshortcutsthataffectsthenavigabilityofgeometricnetworks.Inmostcases,thenumberofshortcutsisproportionaltothecostofthesystem.Therefore,weneedtobeverycarefulwhenplanningthenetwork,duetothepossibilitythatmoreshortcutsmayleadtoworsenetworkperformance.Thisdissertationalsoinvestigatestheproblemofhowmuchbenetnetworkcodingcancontributetothenetworkperformanceintermsofthroughput,delay,andstoragerequirementsformobileadhocnetworks(MANETs),comparedtowhenonlyreplication,storageandforwardingareallowedinrelaynodes.Wecharacterizethethroughput-delay-storagetradeoffsunderdifferentnodemobilitypatterns,i.e.,i.i.d.andrandomwalkmobility,withandwithoutnetworkcoding.OurresultsshowthatwhenrandomlinearcodinginsteadofreplicationisusedinMANETs,anorderimprovementonthescalinglawsofMANETscanbeachieved.Notethatpreviousworkshowedthatnetworkcodingcouldonlyprovideconstantimprovementonthethroughputofstaticwirelessnetworks.OurworkthusdifferentiatesMANETsfromstaticwirelessnetworksbytherolenetworkcodingplays.Foramulti-hopwirelessnetwork(MWN)consistingofmobilenodescontrolledbyindependentself-interestedusers,incentivemechanismisessentialformotivatingmobilenodestocooperateandforwardpacketsforeachother.Existingsolutionssuchasbarterbased,virtual-currencybasedandreputationbasedschemesareeitherless 231

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effectiveorincurhighimplementationcosts,andthereforedonottwellwiththeuniquerequirementsofMWNs.Inthisdissertation,weproposeanovelandpromisingincentiveparadigm,ControlledCodedpacketsasvirtualCommodityCurrency(C4),toinducecooperativebehaviorsinMWNs.InourC4,throughintroducingseveraltechniquesfromnetworkcoding,codedinformationpacketsareutilizedasanewkindofvirtualcurrencytofacilitatepacket/serviceexchangesamongself-interestednodesinaMWN.Sincethevirtualcurrencyimplementedinthiswayalsocarriesusefuldatainformation,itisthecounterpartoftheso-calledcommoditycurrencyinthephysicalworld,andtheoverheadbroughtbyC4isextremelysmallcomparedtotraditionalschemes.WetheoreticallyshowthatC4isperfectlyefcienttosupportMWNswithbroadcastandmulticasttrafcs.Forpureunicastcommunications,byadjustingthegroupingparameter,ourC4providesasystematicwaytosmoothlytradeincentiveeffectivenessforimplementationcost,andtraditionalbarterbasedandvirtual-currencybasedschemesarejusttwoextremecasesofC4.WealsoshowthatwhenourC4iscombinedwiththesocialnetworkformedbymobileusersintheMWN,theimplementationcostscanbefurtherreducedwithoutsacricingincentiveeffectiveness.Inthelastfewyears,trusthasbeenidentiedastheunderlyingandthemosteffectivemechanismtosecureroutingprotocolsinWANETsandaccordinglymanytrustmetricsandtrust-based(orsecure-aware)routingprotocolshavebeenproposedforWANETs.Unfortunately,theissueslikealgebraicpropertiesandcompatibilitiesofthesetrustmetricsarelargelyuntouchedintheliterature.Thecorrectness,optimalityandefciencyofthesetrust-basedroutingprotocolshavebeenanalyzedbyinformalmeansonly.Itiswell-knownthatinformalargumentscanbepronetoerrors,andarescenario-specic.Therefore,thereisastrongneedforamorerigorousandgenerallyapplicableframeworkandtheorytodeepenourunderstandingofthefundamentalrulesgoverningallpossibletrust-basedroutingprotocols,facilitateourformalevaluationandcomparisonsonexistingtrust-basedroutingprotocols,andprovideguidelinesfor 232

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designingnewtrust-basedroutingprotocols.Inthisdissertation,wedevelopaformaltheorytoinvestigatethecorrectness,optimality,andinter-operativityoftrust-basedroutingprotocolsforWANETs.Werstproposeaformalmodeltoabstractthekeyalgebraicpropertiesoftrust-relatedroutingmetricsandidentifythecommonelements,liketrustevaluation,indirecttrustinferenceandtrustworthypathselection,inalltrust-basedroutingprotocols.Next,wedevelopanon-classicalpathalgebrabasedonbi-monoidtostudyindirecttrustinferenceproblems.Wethenprovideasystematicanalysisoftherelationshipbetweentrustmetricsandtrust-basedroutingprotocolsbyidentifyingthebasicalgebraicpropertiesthatatrustmetricmusthaveinordertoworkcorrectlyandoptimallywithdifferentgeneralizeddistance-vectororlink-stateroutingprotocolsinWANETs.Moreover,weextendourframeworktomodeltheinteractionsbetweendifferenttrust-basedroutingprotocols,andcharacterizetheconditionsunderwhichthecorrectnessandoptimalityofroutingoperationscanbeguaranteedinWANETswheremultipleroutingprotocolscoexistordifferenttrustmetricsareadopted.Theproposedresearchprovidesanewmethodologyfortheformalanalysisofwirelessnetworksecurityandacceleratestheevaluation,designandrealdeploymentoftrust-basedroutingprotocols.Securityalwayscomeswithapriceintermsofperformancedegradation,whichshouldbecarefullyquantied.Thisisespeciallythecaseforwirelessadhocnetworks(WANETs)whichoffercommunicationsoverasharedwirelesschannelwithoutanypre-existinginfrastructure.Formingend-to-endsecurepathsinsuchWANETsismorechallengingthaninconventionalnetworksduetothelackofcentralauthorities,anditsimpactonnetworkperformanceislargelyuntouchedintheliterature.Inthisdissertation,basedonageneralrandomnetworkmodel,theasymptoticbehaviorsofsecurethroughputanddelaywiththecommontransmissionrangernandtheprobabilitypfofneighboringnodeshavingaprimarysecurityassociationarequantiedwhenthenetworksizenissufcientlylarge.Thecostsandbenetsofsecure-link-augmentation 233

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operationsonthesecurethroughputanddelayarealsoanalyzed.Ingeneral,securityhasacost:sincewerequireallthecommunicationsoperateonsecurelinks,thereisadegradationinthenetworkperformancewhenpf<1.However,oneimportantexceptionisthatwhenpfis(1=logn),thesecurethroughputremainsattheGuptaandKumarboundof)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(1=p nlognpackets/timeslot,whereinnosecurityrequirementsareenforcedonWANETs.Thisimpliesthatevenwhenthepfgoestozeroasthenetworksizebecomesarbitrarilylarge,itisstillpossibletobuildthroughput-order-optimalsecureWANETs,whichisofpracticalinterestsinceinmanypracticallarge-scaleWANETs,pfisverysmall. 8.2FutureDirectionsTheengineeringandstudyoflarge-scalecomplexnetworkswillbeamajorfocusofscienticresearchinthe21stcentury,particularlyintheareasofcommunications,sociology,biology,andcognitivescience.Italsoposesagreatvarietyofimportantchallengesandopenproblems.Inthefuture,Iwillcontinuetofocusmyresearchondevelopingtheoriesandmethodologiestobetterunderstandthebehavioroflargecomplexnetworks,andengineertheminamoreefcientandsecureway.Mycurrentresearchonwirelessnetworksandnetworksecurityoffersaninitialvalidationofthesemethodologies.Intheshortterm,Iplantoalsoinvestigateothercomplexnetworks,suchasonlinesocialnetworks;andinthelongterm,Iwouldliketoextendmymethodologiestoabroaderscope,thesocalledNetworkScience1,anewandemergingscienticdisciplinethatexaminestheinterconnectionsamongdiversephysicalorengineerednetworks,informationnetworks,biologicalnetworks,cognitiveandsemanticnetworks,andsocialnetworks.Researchinthiseldseekstodiscovercommonprinciples,algorithmsandtoolsthatgovernnetworkbehavior. 1See: http://en.wikipedia.org/wiki/Network_science 234

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SocialNetworkingandItsApplicationsinNetworkingSystemsRecently,manyonlinesocialnetworking(OSN)applicationssuchasFacebook,MySpaceandTwitteremergeasnewwaystoconnectpeople.Theseapplicationsalreadyattractatremendousnumberofusers,andtheircommunitiesareexpectedtogrowmuchlargerinthenearfuture.Manyinterestingquestionsariseunderthenewcontextsoftheseinnovativeapplications.Forexample,whatarethefundamentalcharacteristicsofthesocialgraphsrepresentingusersocialconnections?Howdoesacomputervirusspreadonsuchagraph?Asusersmayloseinterestsinasmallcommunity,whatisthecriticalsizeofausercommunitytoensurethesustainabilityandgrowthofthiscommunity?Iwouldliketoinvestigatetheseproblemsfromatheoreticalperspective,consideringinparticulartheuncertaintyamongsocialconnectionsofusers.Theultimategoalistoderiveusefulinsightstowardsdesigningeffectivealgorithmstoimprovetheutilityofsocialnetworkingapplications.Morespecically,Iamparticularlyinterestedindesigningeffectivelearningalgorithmsbasedonusersocialconnectionstorecommendnewcontactswithcommoninterests,displayadvertisements,andsellproductstointerestedusers,whicharecruciallyimportanttocommercialsuccess.Furthermore,socialnetworkinginformation,asacomplementarychanneltotraditionalnetworkingcharacteristics,ispotentiallyusefultoimprovetheperformanceofothernetworkingsystems.Forexample,mostcurrentdelaytolerantnetworks(DTNs)onlyutilizemobileconnectivityinformationtoselectroutesanddeliverdata.Ontheotherhand,socialnetworkingprovidesindicationsoftheinter-meetingtimeintervalamongusersastheymeetfriendsmorefrequentlythanstrangers.Hence,suchinformationishelpfultofacilitatedataroutinginDTNs.Inaddition,socialconnectionsnaturallyinfertrustsamongfriendsandmaybeusefultoreducesecurityoverhead.Theycanalsohelpdevelopcollaborationeffortsamongdifferentnodestoimproveandoptimizenetworkutilities.Iaminterestedinexploringinnovativewaysandefcient 235

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algorithmstoapplysocialnetworkinginformationtoimproveexistingnetworkingsystems.NetworkScienceandNetworkAlgorithmDesignMyfutureresearchinNetworkSciencewillbefocusedonthefundamentalaspectsofproblemslyingattheheartofalargenumberofcomplexnetworks,fromWANETstobiological,orsocialnetworks.Mystudywillalsobepertinenttothedesignandanalysisofalgorithmsforthesecomplexnetworks.Theoriginalityofmyapproachisbasedonthefollowingmethodologyandconcerns:Firstofall,insteadofwell-knownpropertiesofcomplexnetworks,suchassmallworldorscalefree,mystudywillbebuiltuponthenewdiscoveriesofthenetworkproperties,suchasboundedgrowthrate,lowdoublingdimension,minorexcludingorhyperbolicmetrics.Thesenetworkpropertiesareverynovel.Forinstance,thenotionofdoublingdimensionhasbeenonlyveryrecentlyintroducedwiththeobjectiveoftackinghardproblemssuchasTSP.Notionslikeminorexcluding,orhyperbolicmetricshavebeenseldomlyconsideredforthedesignandanalysisofnetworkalgorithms,whereasitisstronglybelievedthattheyarecriticalinthiscontext.Themainadvantagesoftheaforementionednotionsaretheirgeneralityandpowerfulness.Theyapplytoalargeclassofnetworks,andenableefcientsolutionsforhardproblems.(Forexample,anynetworkwithboundeddoublingdimensionisnavigable,i.e.,greedyroutingperformsinpoly-logarithmicnumberofsteps).Secondly,Iwilltacklesomenetworkproblemsincontextswherethenetworkisonlyimplicitlyand/orpartiallyknown.Theconceptofimplicitknowledgeaimsatcapturingthefactthattheamountofknowledgeeachnodeofadistributednetworkhasaboutanyaspectofthetopology(fromthefullknowledgeofthewholestructureorglobalinformationsuchasnumberofnodes,tolocalinformationsuchasneighborhoodtopology)isinherentlylimited.Implicitknowledgeismodeledbyanoracle,thecomplexityofwhichismeasuredintermsoftheamountofinformation(i.e.,thenumber 236

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ofbits)giventotheentitiesornodescomposingthenetworks.Theconceptofpartialknowledgeaimsatcapturingthefactthatnodesneednotknowalltheirconnections.Thisismodeledbytheprobe-complexityoftheproblems,i.e.,theminimumamountofprobingthatnodesmustperformtosolveaproblem. 237

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APPENDIXANETWORKTOPOLOGIESUSEDINPERFORMANCEEVALUATIONSConsiderthesituationwheresensornodesareindependentlyandrandomlyplacedintheROI.SucharandominitialdeploymentisrequiredwhenindividualsensorplacementisinfeasibleandisdesirablewhenprioriknowledgeoftheROIandthemonitoredtargetislimitedornotavailable.Inthiscases,itiswidelyacceptedintheliterature[ 59 ]thatthelocationsofsensorscanbemodelledbya2-DhomogeneousSpatialPoissonPointProcess(SPPP)withdensity. DenitionA.1. [homogeneousSPPP]AhomogeneousSPPPwithdensitycanbedenedbythefollowingtwoproperties:First,foranymeasurablesubsetofAlwithareaB,PrfndinginodesintheregionofareaBg=(B)ie)]TJ /F28 5.978 Tf 5.76 0 Td[(B i!.Second,thenumberofnodesindisjoint(non-overlapping)areaareindependentrandomvariables.Eachnodeisexpectedtohavek=r2cneighborsonaverage,andtheexpectednumberofnodesinAlisgivenbyn=Al.Wheneachnodefailsindependentlyanduniformlywithprobabilityp.IthasbeenshownthatfunctionalnodesstillformahomogeneousSPPPwithdensity`=(1)]TJ /F3 11.955 Tf 11.95 0 Td[(p)[ 150 ].Inthiscase,thenetworkcanbeuniquelyidentiedbythecurrentnodedensity(orequivalentlyk).Notethatbytuningparameterk(or),wecangetdifferentnetworktopologiescorrespondingtodifferentcoveragepatterns.Foursituationsgetgreatinterestsintheliterature(seeFigure A-1 foranillumination)[ 130 ]:(a)Whenk>4logn+4loglogn,theROIisalmostfullycovered.Therearenoboundarynodeswhenborderinformationisavailable(cf.Figure A-1 (a)).(b)Whenk>logn,thewholenetworkisconnected.Thenumberofinteriornodesislargerthanthatofboundarynodes(cf.Figure A-1 (b)). 238

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FigureA-1. Somenetworktopologiesusedinourperformanceevaluation.(a)k=40.(b)k=15.(c)k=5.(d)k=4.Inallsituations,thepositionoftheBSis(50,50).Shadedarearepresentscoverageofsensors.Noticethatin(c)and(d)onlydarklyshadedarearepresentsconnectedcoverageneededtobemeasured. (c)Whenk>4.5,thenetworkispercolated.Thereonlyonebigcluster,andthenumberofinteriornodesissmallerthanthatofboundarynodes(cf.Figure A-1 (c)).(d)Whenk<4.5,thenetworkissubcritical(collapsed),andconsistsofmanysmallclusters(cf.Figure A-1 (d)).Therefore,inourevaluationwefocusonthecaseswhen 4.5
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Itcanbefoundthat,althoughweonlyusehomogeneousSPPPstogeneratethenetworktopologies,whenthenodedensityissmallerthanthecriticalvaluefortheconnectivity,thenetworktopologieswillbecomeveryirregularduetothedisconnectedness.Inoursimulations,weuseNS-2andassumers=5units,thedatasizeforpositionrepresentationis32bits,theenergyconsumedtotransmitandreceiveonebitis0.8J/bitand0.6J/bit,respectively.SensornodesaredistributedinasquareROIwithsidel=100units.Inthenetworkinitialdeploymentphase,inordertoensurecoverage,totalnumberofnodesdeployedintheROIis1000(correspondingtok=40).TheMACprotocolusedinoursimulationis802.11. 240

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APPENDIXBASYMPTOTICNOTATIONWeusethefollowingstandardnotationthroughoutthedissertation.Fortwononnegativefunctionsf()andg():(i)f(n)=O(g(n))meansthatthereexistsaconstantcandanintegerNsuchthatf(n)cg(n)forn>N(i.e.,asymptoticupperbound);(ii)f(n)=o(g(n))meansthatlimn!1f(n)=g(n)=0(i.e.,asymptoticinsignicance);(iii)f(n)=(g(n))meansthatthereexistsaconstantcandanintegerNsuchthatf(n)cg(n)forn>N(i.e.,asymptoticlowerbound);(iv)f(n)=!(g(n))meansthatlimn!1f(n)=g(n)=1(i.e.,asymptoticdominance);(v)f(n)=(g(n))meansthatf(n)=O(g(n))andg(n)=O(f(n))(i.e.,asymptotictightbound). 241

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APPENDIXCSOMERESULTSABOUTTORUSPARTITIONSINSCHEME1AND3'Asaprelude,werstestablishthefollowingChernoffbound[ 61 ]foraPoissonrandomvariableXofparameter. Lemma15. LetXbeaPoissonrandomvariableofparameter,wehave Pr[Xa]e)]TJ /F19 7.97 Tf 6.59 0 Td[((e)a aa,fora>(C) andPr[Xa]e)]TJ /F19 7.97 Tf 6.58 0 Td[((e)a aa,fora<.(C)For0<<1,ChernoffboundsgiveninEq.( C )andEq.( C )canbecombinedandsimpliedto Pr[jX)]TJ /F7 11.955 Tf 11.96 0 Td[(j]<2e)]TJ /F19 7.97 Tf 6.59 0 Td[(2=2.(C) Proof. NotethatforanyrandomvariableX0,andconstantsa,t0,wehaveXaifandonlyifetXeta.SobyMarkov'sinequality,wehavePr[Xa]EetX eta.ForaPoissonrandomvariableX,wehave EetX=Xk2Netke)]TJ /F19 7.97 Tf 6.58 0 Td[(k k!=e)]TJ /F19 7.97 Tf 6.58 0 Td[(Xk2N(et)k k!=e)]TJ /F19 7.97 Tf 6.58 0 Td[(eet=e(et)]TJ /F9 7.97 Tf 6.58 0 Td[(1).Therefore,wehavePr[Xa]e(et)]TJ /F9 7.97 Tf 6.59 0 Td[(1)e)]TJ /F4 7.97 Tf 6.59 0 Td[(ta=e(et)]TJ /F9 7.97 Tf 6.59 0 Td[(1))]TJ /F4 7.97 Tf 6.59 0 Td[(ta.Fora>,wechooset=log(a=)>0andobtainEq.( C ).Followingasimilarapproach,wecanobtainEq.( C )fora
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Bysubstitutinga=(1)]TJ /F7 11.955 Tf 11.95 0 Td[()intoEq.( C ),weobtain Pr[X(1)]TJ /F7 11.955 Tf 11.95 0 Td[()]e)]TJ /F19 7.97 Tf 6.58 0 Td[( (1)]TJ /F7 11.955 Tf 11.95 0 Td[()(1)]TJ /F19 7.97 Tf 6.59 0 Td[()
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(iii)Consideringanarbitrarycell.Allnodesinthiscellandprimarysecurelinksbetweenthesenodesformasubgraph,whichcanbemodeledasanErdos-Renyirandomgraph[ 16 74 ].Fromtheaboveproof,weknowthatthenumberofnodesineachcellis(logn=pf)andthattheaveragenodedegreeinthissubgraphis(logn),whichislargerthanthelogarithmofthenumberofnodesinthecell,giventhatpf=(logn=n).Therefore,bythepropertiesoftheErdos-Renyirandomgraph[ 16 74 ],thissubgraphisconnected,i.e.,thereexistsasecurepathconnectingarbitrarynodepairsinthecell.Becauseallnodesinthecellareinthetransmissionrangeofeachother,tondthissecurepathonlyneedsone-hoplocalcommunications,whichcanbeignoredcomparedtothemulti-hopdatacommunications. ProofofLemma 12 withPn. (i)BasedonthedescriptionofScheme3`inSection 7.5.1 ,weknowthatwhenpf=o(1=logn),therearem=1=s2n=npf c3squarelets,andthenumberofnodesineachsquareletisaPoissonrandomvariableXofparameter=ns2n=c3=pf,wherec3isaconstantandpf=(logn=n)andpf=o(1=logn).For0<<1,letAnbetheeventthatthereisatleastonesquareletwithmorethan(1+)orlessthan(1)]TJ /F7 11.955 Tf 11.96 0 Td[()nodes.BytheunionboundandEq.( C )inLemma 15 ,wehave Pr[An]mPr[jX)]TJ /F7 11.955 Tf 11.96 0 Td[(j]<2npf c31 nc32 2!0asntendstoinnityforanyc34=2.Therefore,eachsquareletcontains()=(ns2n)nodesw.h.p.Whenpf=(1=logn),theproofofLemma 12 (i)isstraightforwardandisomittedhereduetospaceconstraints.(ii)SupposethereareZnodesinthenetwork,whereZisaPoissonrandomvariableofparametern.Fromourpreviouswork[ 168 ],weknowthat(1)]TJ /F3 11.955 Tf 12.05 .01 Td[(pi)jZjnodes(calledbackbonenodes)areconnectedbyasecurebackbone(alsocalledthegiantclusterinthepercolationliterature)withprimarysecurelinks.TherearestillpijZj 244

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nodes(calledisolatednodes)disconnectedfromthegiantcluster,where01)]TJ /F15 11.955 Tf 11.96 16.86 Td[(1)]TJ /F3 11.955 Tf 19.7 8.09 Td[(c4 logn(1)]TJ /F19 7.97 Tf 6.59 0 Td[()c3logn>1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F9 7.97 Tf 6.59 0 Td[((1)]TJ /F19 7.97 Tf 6.58 0 Td[()c3c4,where,c3andc4areallconstants.Whenpf=o(1=logn),thenumberofnodesineachsquarelet,i.e.,jXj,islowerboundedby(1)]TJ /F7 11.955 Tf 11.95 0 Td[()c3=pf.Wethushavepl=1)]TJ /F6 11.955 Tf 11.95 0 Td[((1)]TJ /F3 11.955 Tf 11.95 0 Td[(pf)jXj>1)]TJ /F6 11.955 Tf 11.96 -.17 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(pf)(1)]TJ /F19 7.97 Tf 6.59 0 Td[()c3=pf>1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F9 7.97 Tf 6.59 0 Td[((1)]TJ /F19 7.97 Tf 6.59 0 Td[()c3,whereandc3areallconstants.(iv)Firstly,fromEq.( 7 )inScheme3`wedirectlyarriveattheconclusionthateachsquareletcontainsatleastonenodeonthesecurebackbonew.h.p.Sincern=p 5sn,weknowthatinnodei'stransmissionrange,thereexistsatleastonesquarelet.Therefore,thereexistsatleastonenodeinnodei'stransmissionrangewhichbelongstothesecurebackbonew.h.p.Secondly,recallthatinScheme3`wecanguaranteethattheprimarysecuregraphisinthepercolatedphasew.h.p..Wealsohaveproved 245

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FigureC-1. Cellschedulingscheme.HereisanillustrationofthecellsbeingdividedintoK2groupsforthecaseofK=3,i.e.,9groups.Allthebluecellswhichareingroup1transmitinthesametimeslot.Inthenexttimeslotallthecellsingroup2transmitandsoon. inourpreviouswork(cf.Theorem2in[ 168 ])thatwhentheprimarysecuregraphisinthepercolatedphase,eachnodebelongstothesecurebackbonewithaprobabilityS,whereSisaconstant.Whenpf=(logn=n),thereareatleast(logn)friendsofnodeiinthewholenetwork,andeachfriendbelongstothesecurebackbonewiththeprobabilityS.FromtheChernoffbound,itiseasytoshowthatatleastoneofthese(logn)friendsbelongstothesecurebackbone. NotethatXncanbewellapproximatedbyPnasntendstoinnity.Therefore,bythede-Poissonizationtechniqueintroducedin[ 130 ,Section2.5,p.37],wecanprovethatLemma 10 andLemma 12 alsoholdwhennodesfollowauniformpointprocess,i.e.,Xn,forntendingtoinnity.Duetospaceconstraints,weomitthisroutineproofhere. 246

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APPENDIXDSECURENETWORKPERFORMANCEUNDERTHEPHYSICALMODELHereweshowthatthesameresultsonsecureWANETsasinTheorems 7.1 and 7.2 canbeobtainedunderthephysicalmodel.WerstshowthattheconstructivelowerboundsprovidedinSection 7.4 and 7.5 willnotbechangedunderthephysicalmodel.NotethattheprotocolmodelonlyrelatestothecellschedulingpartofschemesproposedinSection 7.4 and 7.5 .Therefore,ifwecanshowthatthesamepropertyofthecellschedulingasdescribedinProposition 7.1 stillholdsforthephysicalmodel,wearedone.Inwhatfollows,weprovethisclaimbasedontheassumptionthat>2. ProofofProposition 7.1 underthephysicalmodel. WeusethesamecellschedulingschemeasinProposition 7.1 undertheprotocolmodel(seeFigure C-1 foranillustration).ThereceivedpowerofthedesiredsignalislowerboundedbyPnGij=Pnd)]TJ /F19 7.97 Tf 6.59 0 Td[(ijPn(p 5cn))]TJ /F19 7.97 Tf 6.59 0 Td[(,wherecnisthesidelengthofeachcell.Wethenboundtheinterference,i.e.,I.ConsideraparticularcellC.Ifonenodefromthiscellistransmitting,allothersimultaneoustransmissionsmayoccurincellsbelongingtothesamesetofcellsthatareasaverticalandhorizontaldistanceofexactlysomemultiplesofaparticularintegerK.Actually,theinterferingcellsareplacedalongtheperimeterofconcentricsquares,whosecenterisC,andeachsquarecontains2Ki(i=1,2,,L)interferingcellsasdepictedinFigure C-1 ,whereListhenumberofsuchconcentricsquares.Forexample,therstconcentricsquarecontains8interferingcells,whereasthesecondconcentricsquarecontains16interferingcells,fortheparticularcasewhereK=4.EachnodeintheintendedcellCtransmitsdatapacketstonodesinthefourneighboringcells.Then,thedistancebetweenthesenodes(thepossiblereceiversinthefouradjacentcells)andtheinterferingonesisat 247

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least(K)]TJ /F6 11.955 Tf 12.64 0 Td[(2)cni(i=1,2,,L).Asweareconsideringalowerbound,wetaketheworstcase.Then,thenumberofconcentricsquares(irrespectiveofthepositionoftheintendedcell,becausetheworstcaseiswhentheintendedcellisatonecornerofthearea)isatmostLl1 2Kcnm.Weproceedtoupper-boundtheinterferenceatthereceiverjasI=Xk2SnfjgPnd)]TJ /F19 7.97 Tf 6.58 0 Td[(kj(RecallthatSistheconcurrenttransmitterset)LXi=1Pn2Ki [(K)]TJ /F6 11.955 Tf 11.95 0 Td[(2)cni]=2PnK [(K)]TJ /F6 11.955 Tf 11.95 0 Td[(2)cn]LXi=1i1)]TJ /F19 7.97 Tf 6.59 0 Td[(2PnK [(K)]TJ /F6 11.955 Tf 11.95 0 Td[(2)cn]1+ZL1x1)]TJ /F19 7.97 Tf 6.59 0 Td[(dx=2PnK [(K)]TJ /F6 11.955 Tf 11.95 0 Td[(2)cn]1+1 2)]TJ /F7 11.955 Tf 11.96 0 Td[((L2)]TJ /F19 7.97 Tf 6.58 0 Td[()]TJ /F6 11.955 Tf 11.95 0 Td[(1)=2PnK [(K)]TJ /F6 11.955 Tf 11.95 0 Td[(2)cn])]TJ /F6 11.955 Tf 11.96 0 Td[(1 )]TJ /F6 11.955 Tf 11.96 0 Td[(2+2PnK [(K)]TJ /F6 11.955 Tf 11.96 0 Td[(2)cn]L2)]TJ /F19 7.97 Tf 6.59 0 Td[( 2)]TJ /F7 11.955 Tf 11.96 0 Td[(c5PnK [(K)]TJ /F6 11.955 Tf 11.95 0 Td[(2)cn](Recallthat>2),wherec5isapositiveconstant.Therefore,basedonthephysicalmodel(cf.Section 7.3.1.2 ),wehave SINRijPn(p 5cn))]TJ /F19 7.97 Tf 6.59 0 Td[( N0+c5PnK [(K)]TJ /F9 7.97 Tf 6.59 0 Td[(2)cn]=c6Pn c7cnN0+c8Pn,(D)wherec6,c7andc8areconstants.Recallthatcn1.Therefore,SINRijinEq.( D )canbelower-boundedbysomeconstant,whichguaranteesthesuccessfulreceptionofpacketsatnodej.SowecompletetheproofthatProposition 7.1 alsoholdsunderthephysicalmodel. Next,weshowthattheupperboundonthesecurethroughputandthelowerboundonthee2edelayprovidedinSection 7.6 willnotbechangedunderthephysicalmodel. 248

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Notethattheinterferencemodelonlyaffectstheinterferenceconstraint.Therefore,ifwecanshowthatthephysicalmodelyieldsthesameinterferenceconstraint,wearedone.Thefollowinglemmaontheexistenceofacorrespondencebetweenphysicalandprotocolmodelsonsimultaneoustransmissionsetsguaranteesthatitisindeedthecase. Lemma16. Let()=482)]TJ /F23 5.978 Tf 5.76 0 Td[(2 )]TJ /F9 7.97 Tf 6.58 0 Td[(21=.Supposethatfor>()theprotocolmodelallowssimultaneoustransmissionsforatransmitter-receiver(T-R)pairinasetS.ThenthereexistsapowerassignmentfPi,1ingallowingthesameT-SpairsetSunderthephysicalmodelwiththreshold. Proof. cf.theproofofTheorem4.1in[ 163 ,p.174]. 249

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BIOGRAPHICALSKETCH ChiZhangwasbornin1977inWuhan,Hubei,China.Theolderoftwochildren,ChigrewupinWuhanandgraduatedfromtheNO.1MiddleSchoolafliatedtoCentralChinaNormalUniversityinthesummerof1995.Followinghighschool,ChienrolledatHuazhongUniversityofScienceandTechnology(HUST)inWuhan,Chinainthefallof1995.HereceivedhisB.E.andM.E.degreesinelectricalandinformationengineeringfromHUST,in1999and2002,respectively.ChienrolledinthePh.D.programintheDepartmentofElectricalandComputerEngineeringattheUniversityofFloridainthefallof2004,asarecipientoftheUniversityofFlorida'sAlumniFellowship.HereceivedhisPh.D.degreeinelectricalandcomputerengineeringfromtheUniversityofFloridainthesummerof2011.Hisresearchinterestsareintheareasofnetworkprotocoldesign,networkperformanceanalysis,andnetworksecurityguarantee,particularlyforwirelessnetworksandsocialnetworks.Hehaspublishedover50papersinprestigiousjournalsincludingIEEE/ACMTransactionsonNetworking,IEEEJournalonSelectedAreasinCommunications,andIEEETransactionsonMobileComputing,orintopnetworkingconferencessuchasIEEEINFOCOM,IEEEICNP,andIEEEICDCS.HehasservedastheTechnicalProgramCommittee(TPC)membersforseveralinternationalconferencesincludingIEEEGLOBECOM2010and2011,IEEEICC2011,andIEEEPIMRC2011.Chiisthe2011recipientoftheGatorEngineerGraduateStudentAttributeAwardforCreativity(onerecipientperyear).ChialsohasbeenselectedthreetimesasarecipientofthetravelgrantfromtheNationalScienceFoundation(NSF)toattendIEEEICDCS2008,IEEEINFOCOM2009and2011.ChiisastudentmemberofIEEE. 264