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The Exploitation of Power-Law Networks

Permanent Link: http://ufdc.ufl.edu/UFE0045394/00001

Material Information

Title: The Exploitation of Power-Law Networks Robustness, Optimization and Its Impact on Communication Networks and Social Behaviors
Physical Description: 1 online resource (119 p.)
Language: english
Creator: Shen, Yilin
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: approximationalgorithm -- communicationnetwork -- hardnessandinapproxmibability -- networkvulnerability -- optimization -- powerlawgraph -- socialnetwork
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Many practical complex networks, such as the Internet, WWW and social networks, are discovered to follow power-law distribution in their degree sequences, i.e., the number of nodes with degree $i$ in these networks is proportional to $i^{-\beta}$ for some exponential factor $\beta > 1$. The exploitation of such networks becomes an urgent need, yet remains open especially from theoretical viewpoints. In this dissertation, we first investigate if it is easier to solve many optimization problems in power-law networks. Our works focus on the hardness and inapproximability of optimization problems on power-law graphs (PLG). Particularly, we show that the \textsc{Minimum Dominating Set}, \textsc{Minimum Vertex Cover} and \textsc{Maximum Independent Set} are still {\em APX}-hard on power-law graphs. We further show the inapproximability factors of these optimization problems and a more general problem ($\rho$-\textsc{Minimum Dominating Set}), which proved that a belief of $(1 + o(1))$-approximation algorithm for these problems on power-law graphs is not always true. In order to show the above theoretical results, we propose a general cycle-based embedding technique to embed any $d$-bounded graphs into a power-law graph. In addition, we present a brief description of the relationship between the exponential factor $\beta$ and constant greedy approximation algorithms. Moreover, we propose a algorithm framework, called Low-Degree Percolation (LDP) Algorithm Framework, for solving Minimum Dominating Set, Minimum Vertex Cover and Maximum Independent Set problems in power-law graphs. Using this framework, we further show a theoretical framework to derive the approximation ratios for these optimization problems in two well-known random power-law graphs. Numerical analysis shows that our proposed framework can not only lead to a good theoretical approximation ratio but also result in even better performance than theoretical bounds. In addition, the robustness of power-law networks attracts more research attentions since they are exposed to a great number of threats such as adversarial attacks on the Internet, cybercrimes on the WWW or malware propagations on social networks. In this dissertation, we first show it NP-hard to detect critical links and nodes even in power-law networks. Due to the denial of promptly assessing vulnerability of power-law networks in this manner, we are more interested in the vulnerability of power-law networks under random attacks and adversarial attacks using the in-depth probabilistic analysis on the theory of random power-law graph models. Our results indicate that power-law networks are able to tolerate random failures if their exponential factor $\beta$ is less than $2.9$, and they are more robust against intentional attacks if $\beta$ is smaller. In the present of cascading failure, we show that power-law networks are very vulnerable when cascading failure occurs since any random failures of high degree nodes can easily overload the low degree nodes. At last, we study the optimization of power-law networks, from design and protection perspectives. On the one hand, we reveal the best range $1.8, 2.5$ for the exponential factor $\beta$ by optimizing the complex networks in terms of both their vulnerabilities and costs. When $\beta 2.5$ the network robustness is unpredictable since it depends on the specific attacking strategy. On the other hand, we study \emph{Critical Link Disruptor (CLD)} and \emph{Critical Node Disruptor (CND)} optimization problems to identify critical links and nodes in a network whose removals maximally destroy the network's functions. After showing the NP-hardness of these two problems, we propose HILPR, a novel LP-based rounding algorithm, for efficiently solving CLD and CND problems in a timely manner. In the case of cascading failures, we further develop the TRPA algorithm, an iterative 2-phase algorithm, for solving \emph{Cascading Critical Node Disruptor (CCND)} problem. The effectiveness of our solutions is validated on various synthetic and real-world networks.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: 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 Yilin Shen.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Thai, My Tra.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-05-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045394:00001

Permanent Link: http://ufdc.ufl.edu/UFE0045394/00001

Material Information

Title: The Exploitation of Power-Law Networks Robustness, Optimization and Its Impact on Communication Networks and Social Behaviors
Physical Description: 1 online resource (119 p.)
Language: english
Creator: Shen, Yilin
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: approximationalgorithm -- communicationnetwork -- hardnessandinapproxmibability -- networkvulnerability -- optimization -- powerlawgraph -- socialnetwork
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Many practical complex networks, such as the Internet, WWW and social networks, are discovered to follow power-law distribution in their degree sequences, i.e., the number of nodes with degree $i$ in these networks is proportional to $i^{-\beta}$ for some exponential factor $\beta > 1$. The exploitation of such networks becomes an urgent need, yet remains open especially from theoretical viewpoints. In this dissertation, we first investigate if it is easier to solve many optimization problems in power-law networks. Our works focus on the hardness and inapproximability of optimization problems on power-law graphs (PLG). Particularly, we show that the \textsc{Minimum Dominating Set}, \textsc{Minimum Vertex Cover} and \textsc{Maximum Independent Set} are still {\em APX}-hard on power-law graphs. We further show the inapproximability factors of these optimization problems and a more general problem ($\rho$-\textsc{Minimum Dominating Set}), which proved that a belief of $(1 + o(1))$-approximation algorithm for these problems on power-law graphs is not always true. In order to show the above theoretical results, we propose a general cycle-based embedding technique to embed any $d$-bounded graphs into a power-law graph. In addition, we present a brief description of the relationship between the exponential factor $\beta$ and constant greedy approximation algorithms. Moreover, we propose a algorithm framework, called Low-Degree Percolation (LDP) Algorithm Framework, for solving Minimum Dominating Set, Minimum Vertex Cover and Maximum Independent Set problems in power-law graphs. Using this framework, we further show a theoretical framework to derive the approximation ratios for these optimization problems in two well-known random power-law graphs. Numerical analysis shows that our proposed framework can not only lead to a good theoretical approximation ratio but also result in even better performance than theoretical bounds. In addition, the robustness of power-law networks attracts more research attentions since they are exposed to a great number of threats such as adversarial attacks on the Internet, cybercrimes on the WWW or malware propagations on social networks. In this dissertation, we first show it NP-hard to detect critical links and nodes even in power-law networks. Due to the denial of promptly assessing vulnerability of power-law networks in this manner, we are more interested in the vulnerability of power-law networks under random attacks and adversarial attacks using the in-depth probabilistic analysis on the theory of random power-law graph models. Our results indicate that power-law networks are able to tolerate random failures if their exponential factor $\beta$ is less than $2.9$, and they are more robust against intentional attacks if $\beta$ is smaller. In the present of cascading failure, we show that power-law networks are very vulnerable when cascading failure occurs since any random failures of high degree nodes can easily overload the low degree nodes. At last, we study the optimization of power-law networks, from design and protection perspectives. On the one hand, we reveal the best range $1.8, 2.5$ for the exponential factor $\beta$ by optimizing the complex networks in terms of both their vulnerabilities and costs. When $\beta 2.5$ the network robustness is unpredictable since it depends on the specific attacking strategy. On the other hand, we study \emph{Critical Link Disruptor (CLD)} and \emph{Critical Node Disruptor (CND)} optimization problems to identify critical links and nodes in a network whose removals maximally destroy the network's functions. After showing the NP-hardness of these two problems, we propose HILPR, a novel LP-based rounding algorithm, for efficiently solving CLD and CND problems in a timely manner. In the case of cascading failures, we further develop the TRPA algorithm, an iterative 2-phase algorithm, for solving \emph{Cascading Critical Node Disruptor (CCND)} problem. The effectiveness of our solutions is validated on various synthetic and real-world networks.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: 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 Yilin Shen.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Thai, My Tra.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-05-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045394:00001


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THEEXPLOITATIONOFPOWER-LAWNETWORKS:ROBUSTNESS,OPTIMIZATIONANDITSIMPACTONCOMMUNICATIONNETWORKSANDSOCIALBEHAVIORSByYILINSHENADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013YilinShen 2

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Idedicatethistomyparentsandmygirlfriend. 3

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ACKNOWLEDGMENTS IwouldliketotakethisopportunitytothankmycommitteechairDr.ThaiforherpricelesshelpformyPh.D.program.NotonlythechancethatsheofferedmetofurthermystudyandresearchonComputerNetworkingformyPh.D.degree,butalsoherpreciousguidanceovermy4-yearPh.D.studyandresearch,areindispensabletothisthesis.Herstrongpassion,precisenessandprofoundknowledgeforresearchhavebeenenlighteningmethroughoutmyPh.D.period.ThenancialsupportfromherevadesmefromthenancialproblemsforinternationalstudentsandsothatIcanconcentrateovertheresearches.Iwouldalsoliketothankalltheprofessors,Prof.SartajSahni,Prof.SanjayRanka,Prof.PrabhatMishraandProf.PanosM.Pardalos,inmycommitteefortheirtimefordiscussingovermyresearchtopicsandprovidingnumerousconstructiveopinions.Iwouldliketothankmygroupmembers,YingXuan,ThangN.Dinh,NamP.Nguyen,DzungT.Nguyen,RaviTiwari,IncheolShin,HuiyuanZhangfortheirhelpinmystudyandwork.MyresearchwaspartiallyfundedbyDTRA,YoungInvestigatorAward,BasicResearchProgram#HDTRA1-09-1-0061andDTRA#HDTRA1-08-10. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 14 1.1Power-LawGraphs ............................... 14 1.1.1FormalDenition ............................ 14 1.1.2RandomPower-LawGraphModel .................. 14 1.2OptimizationProblemsinPower-LawGraphs ................ 15 1.3VulnerabilityAssessmentofPower-LawNetworks .............. 17 1.4OptimizationofPower-LawNetworks ..................... 18 1.5OutlineofDissertation ............................. 18 2HARDNESSANDAPPROXIMATIONALGORITHMS ............... 20 2.1Preliminaries .................................. 20 2.1.1ProblemDenitions ........................... 21 2.1.2SomeNotations ............................. 22 2.1.3SpecialGraphs ............................. 22 2.1.4ExistingInapproximabilityResults ................... 23 2.2InapproximabilityOptimalSubstructureFrameworkinPower-LawGraphs 24 2.3HardnessandInapproximabilityofOptimalSubstructureProblems .... 25 2.3.1GeneralCycle-BasedEmbeddingTechnique ............ 25 2.3.2APX-Hardness ............................. 27 2.3.3InapproximabilityFactors ........................ 29 2.4MoreInapproximabilityResultsonSimplePower-LawGraphs ....... 34 2.4.1GeneralGraphicEmbeddingTechnique ............... 34 2.4.2InapproximabilityofMIS,MVCandMDS ............... 36 2.4.3MaximumClique,MinimumColoring ................. 38 2.5RelationshipbetweenandApproximationHardness ........... 39 2.6MinorNP-HardnessonSimplePower-LawGraphsfor<1 ........ 40 2.7ApproximationAlgorithms ........................... 42 2.7.1Low-DegreePercolation(LDP)AlgorithmFramework ........ 43 2.7.2ApproximationRatioAnalysis ..................... 44 2.7.2.1Theoreticalframework .................... 44 2.7.2.2Power-lawrandomgraph .................. 48 2.8RelatedWorks ................................. 51 5

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3VULNERABILITYASSESSMENT ......................... 53 3.1Metric ...................................... 53 3.2ThreatTaxonomyandNotations ....................... 55 3.2.1ThreatTaxonomy ............................ 55 3.2.2NotationExplanation .......................... 56 3.3Preliminaries .................................. 56 3.3.1PreviousWorks ............................. 56 3.3.2RobustnessofIntactPower-lawNetworks .............. 57 3.4RandomFailures ................................ 58 3.4.1RobustnessunderRandomFailures ................. 58 3.4.2GoodRangeofunderRandomFailures .............. 62 3.5PreferentialAttacks ............................... 63 3.5.1InteractivePreferentialAttackspi=1)]TJ /F9 7.97 Tf 16.11 4.71 Td[(1 i0 ............ 63 3.5.2ExpectedPreferentialAttackspi=ci e()]TJ /F9 7.97 Tf 6.58 0 Td[(1) ........... 65 3.5.3RelationsbetweenandExpectedAttackedNodes ........ 66 3.6Degree-CentralityAttacks ........................... 67 3.6.1RobustnessunderDegree-CentralityAttacks ............ 68 3.6.2RelationsbetweenandAttackedNodes .............. 69 3.7RandomCascadingFailures ......................... 70 3.7.1CascadingFailureModel ........................ 70 3.7.2CascadingRandomFailures ...................... 72 3.7.3NumericalAnalysis ........................... 77 3.8RelatedWorks ................................. 78 4OPTIMIZATIONOFPOWER-LAWNETWORKS ................. 80 4.1DesignOptimizationofPower-lawNetworks ................. 80 4.1.1CommunicationNetworks ....................... 81 4.1.2SocialNetworks ............................. 83 4.1.3OptimalRangeofExponentialFactor ................ 84 4.2CriticalElementsDetectioninPower-lawNetworks ............. 86 4.2.1HardnessofDetectingCriticalLinksandNodes ........... 86 4.2.2HILPRApproach ............................ 89 4.2.2.1Integerlinearprogrammingformulation .......... 90 4.2.2.2Hybriditerativelproundingalgorithm ........... 91 4.2.2.3Performanceevaluation ................... 94 4.2.3TRGAApproachunderCascadingFailures ............. 100 4.2.3.1TRGA:aniterative2-phasealgorithm ........... 100 4.2.3.2OptimalityofCCNDproblem ................ 102 4.2.3.3Experimentalevaluation ................... 103 4.3RelatedWorks ................................. 106 5CONCLUSION .................................... 109 6

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REFERENCES ....................................... 111 BIOGRAPHICALSKETCH ................................ 119 7

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LISTOFTABLES Table page 2-1InapproximabilityFactorsonPower-LawGraphswithExponentialFactor>1 20 8

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LISTOFFIGURES Figure page 2-1SpecialGraphExamples:Theleftoneisa(3,3,3,3,3,3,3,3)-regularcycleandtherightoneisa(3,3,3,3)-branch-(2,2,2,2,2,2)-cycle.ThegreyverticesconsistoftheoptimalsolutionofMDSonthesetwospecialgraphs. ...... 23 2-2TheReductionfromMVCto-MDS ........................ 32 2-3NumericalresultsofourLDPalgorithmsondifferent(=5):(1)Theoreticalresultsshowstheapproximationratioswithprobabilityatleast1)]TJ /F7 11.955 Tf 12.91 0 Td[(o(1).Asonecansee,ourLDPalgorithmscanobtaintheoptimalsolutionforalltheseproblemsaftergetslargerthan1.6and1.7inERPLandSRPLrespectively,whichcoverstherangeofinmostreal-worldnetworks[ 18 ].Fortheothersmallerexponentialfactors,wecanseethattheapproximationratiosarealittlebithigher,especiallyupto5forMDSandMISproblemsforSRPLmodel.However,theprobabilitiesthatthesetwoproblemscanobtaintheapproximationratioslessthan1.5usingLDPalgorithmsareatleast0.95(onlyalittlebitlowerthan1)]TJ /F7 11.955 Tf 9.3 0 Td[(o(1)).(2)ExperimentalresultsfurtherrevealsthatourLDPalgorithmscanachieveevenbettersolutionsthantheoreticalbounds.(Wetestson100casesandchoosetheaverage.)AsillustratedinFig. 2-3 ,theapproximationratiosofallMDS,MVC,MISproblemsisnolargerthan1.2and2.5evenwhen=1.3inERPLandSRPLmodelsrespectively. ................. 51 3-1AnExampleofInternet:theremovalofv8andv10(greynodes)issufcienttodestroythefunctionofthewholenetworksuchthatonlylessthan40%nodesconnecteachother. ................................. 54 3-2RelationbetweenThresholdpandFailureProbabilityp ............ 62 3-3RelationbetweenandAttackedNodesunderIterativePreferentialAttacks 67 3-4RelationbetweenandAttackedNodesunderExpectedPreferentialAttacks 67 3-5RelationbetweenandAttackedNodesunderDegree-CentralityAttacks ... 69 3-6Eachnodeinthispowergridhasloadequaltoitsdegree,capacityequaltotwiceitsdegreeandeachredarrowsaystheshiftingof2unitload.Thesolidredarrowsstandforthedirectfailurecausedbythecascadesandthedottedonesmeantheloadshiftingtotheneighborwhichisnotfaileddirectly.Theoverloadandfailureofv8andv10canonlycausethedisconnectionfromgeneratorsandtransmitters,yetthepowercanbestillsuppliedtocustomersfromdemandcenters.However,whenfailurecascades,itleadstothebreakdownofalltransmittersandtheelectricitytocustomersareaffectedinstantly. .............. 71 9

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3-7NumericalAnalysisinPower-LawNetworks(=1.5,n=250).Weplotthethreecascadinghopsandndthatouranalysis(pinkplots)approximatesthesimulationofthetotalpairwiseconnectivity(PWC)aftercascadingfailuressurprisinglywell,inbothcasesthatpower-lawnetworksarea.s.unaffected(PWC/n2)anda.s.fragmented. .......................... 78 4-1OptimalRobustCommunicationNetworks .................... 85 4-2OptimalRobustSocialNetworks .......................... 85 4-3AnexampleofCNDreductiononPLGs.Forsimplicity,wejustdrawthenodesinGanditsnewlyaddednodesandlinks. ..................... 89 4-4Triangleinequalityconstraints ........................... 93 4-5TheperformanceofHILPRusingdifferentinterroristnetwork ......... 95 4-6TheperformanceevaluationofHILPRagainstthedegreeandbetweennesscentralityalgorithmsfortheCLDproblem ..................... 96 4-7TheperformanceevaluationofHILPRagainstthedegreeandbetweennesscentrality,andCNLSalgorithmsfortheCNDproblem ............... 96 4-8OverlappingcriticalnodesbetweenoptimalsolutionandHILPRinterroristnetwork ........................................ 97 4-9Thecomparisonofdifferentmetricsonterroristnetwork ............. 99 4-10TheperformanceevaluationofTRGAagainstdegreeandbetweennesscentralityalgorithmsfortheCCNDproblem .......................... 105 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyTHEEXPLOITATIONOFPOWER-LAWNETWORKS:ROBUSTNESS,OPTIMIZATIONANDITSIMPACTONCOMMUNICATIONNETWORKSANDSOCIALBEHAVIORSByYilinShenMay2013Chair:MyT.ThaiMajor:ComputerEngineeringManypracticalcomplexnetworks,suchastheInternet,WWWandsocialnetworks,arediscoveredtofollowpower-lawdistributionintheirdegreesequences,i.e.,thenumberofnodeswithdegreeiinthesenetworksisproportionaltoi)]TJ /F12 7.97 Tf 6.58 0 Td[(forsomeexponentialfactor>1.Theexploitationofsuchnetworksbecomesanurgentneed,yetremainsopenespeciallyfromtheoreticalviewpoints.Inthisdissertation,werstinvestigateifitiseasiertosolvemanyoptimizationproblemsinpower-lawnetworks.Ourworksfocusonthehardnessandinapproximabilityofoptimizationproblemsonpower-lawgraphs(PLG).Particularly,weshowthattheMINIMUMDOMINATINGSET,MINIMUMVERTEXCOVERandMAXIMUMINDEPENDENTSETarestillAPX-hardonpower-lawgraphs.Wefurthershowtheinapproximabilityfactorsoftheseoptimizationproblemsandamoregeneralproblem(-MINIMUMDOMINATINGSET),whichprovedthatabeliefof(1+o(1))-approximationalgorithmfortheseproblemsonpower-lawgraphsisnotalwaystrue.Inordertoshowtheabovetheoreticalresults,weproposeageneralcycle-basedembeddingtechniquetoembedanyd-boundedgraphsintoapower-lawgraph.Inaddition,wepresentabriefdescriptionoftherelationshipbetweentheexponentialfactorandconstantgreedyapproximationalgorithms.Moreover,weproposeaalgorithmframework,calledLow-DegreePercolation(LDP)AlgorithmFramework,forsolvingMinimumDominatingSet,MinimumVertexCoverandMaximumIndependentSetproblemsinpower-law 11

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graphs.Usingthisframework,wefurthershowatheoreticalframeworktoderivetheapproximationratiosfortheseoptimizationproblemsintwowell-knownrandompower-lawgraphs.Numericalanalysisshowsthatourproposedframeworkcannotonlyleadtoagoodtheoreticalapproximationratiobutalsoresultinevenbetterperformancethantheoreticalbounds.Inaddition,therobustnessofpower-lawnetworksattractsmoreresearchattentionssincetheyareexposedtoagreatnumberofthreatssuchasadversarialattacksontheInternet,cybercrimesontheWWWormalwarepropagationsonsocialnetworks.Inthisdissertation,werstshowitNP-hardtodetectcriticallinksandnodeseveninpower-lawnetworks.Duetothedenialofpromptlyassessingvulnerabilityofpower-lawnetworksinthismanner,wearemoreinterestedinthevulnerabilityofpower-lawnetworksunderrandomattacksandadversarialattacksusingthein-depthprobabilisticanalysisonthetheoryofrandompower-lawgraphmodels.Ourresultsindicatethatpower-lawnetworksareabletotoleraterandomfailuresiftheirexponentialfactorislessthan2.9,andtheyaremorerobustagainstintentionalattacksifissmaller.Inthepresentofcascadingfailure,weshowthatpower-lawnetworksareveryvulnerablewhencascadingfailureoccurssinceanyrandomfailuresofhighdegreenodescaneasilyoverloadthelowdegreenodes.Atlast,westudytheoptimizationofpower-lawnetworks,fromdesignandprotectionperspectives.Ontheonehand,werevealthebestrange[1.8,2.5]fortheexponentialfactorbyoptimizingthecomplexnetworksintermsofboththeirvulnerabilitiesandcosts.When<1.8,thenetworkmaintenancecostisveryexpensive,andwhen>2.5thenetworkrobustnessisunpredictablesinceitdependsonthespecicattackingstrategy.Ontheotherhand,westudyCriticalLinkDisruptor(CLD)andCriticalNodeDisruptor(CND)optimizationproblemstoidentifycriticallinksandnodesinanetworkwhoseremovalsmaximallydestroythenetwork'sfunctions.AftershowingtheNP-hardnessofthesetwoproblems,weproposeHILPR,anovelLP-based 12

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roundingalgorithm,forefcientlysolvingCLDandCNDproblemsinatimelymanner.Inthecaseofcascadingfailures,wefurtherdeveloptheTRPAalgorithm,aniterative2-phasealgorithm,forsolvingCascadingCriticalNodeDisruptor(CCND)problem.Theeffectivenessofoursolutionsisvalidatedonvarioussyntheticandreal-worldnetworks. 13

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CHAPTER1INTRODUCTIONOneofthemostremarkablediscoveriesinmanyreal-worldnetworksisthepower-lawdistributionintheirdegreesequences,rangingfromtheInternet[ 34 ],WWW[ 4 ],biologicalnetworks[ 12 ]tosocialnetworks[ 74 ].Inparticular,thenumberofnodeswithdegreeiinthesecomplexnetworksisobservedtobeproportionaltoi)]TJ /F12 7.97 Tf 6.59 0 Td[(forsomeexponentialfactor>1. 1.1Power-LawGraphs 1.1.1FormalDenitionWeconsiderthefollowinggraph,(,)graphG(,),withitspower-lawdegreedistributiondependingontwogivenvaluesand. Denition1((,)GraphG(,)). GivenanundirectedgraphG=(V,E)havingjVj=nnodesandjEj=medges,itiscalleda(,)power-lawgraphifitsmaximumdegreeis=e=andthenumberofnodeswithdegreeiis yi=8><>:be ic,ifi>1orPi=1be icisevenbec+1,otherwise(1)Notethatthenumberofnodesn=e()+O(n1 )]TJ /F5 11.955 Tf 12.86 0 Td[(1)andthenumberofedgesm=1 2e()]TJ /F5 11.955 Tf 12.55 0 Td[(1)+O(n2 )]TJ /F5 11.955 Tf 12.55 0 Td[(1),where()=P1i=11 iistheRiemannZetafunction.Forsimplicity,sincethereisonlyaverysmallerroro(1)when>2whencountingthenumberofbothnodesandedges,wedenotethemasn.=e()andedgesm.=1 2e()]TJ /F5 11.955 Tf 11.95 0 Td[(1). 1.1.2RandomPower-LawGraphModelTherearetwomaincategoriesofrandomgraphmodelstogenerategraphswithskeweddegreesequences,evolutionaryandstructural.Evolutionarymodelsleadtotheskeweddegreedistributionsbyidentifyinggrowthprimitives,includingmulti-objectiveoptimization[ 6 33 ]andstatisticalpreferentialattachment[ 11 24 58 65 ].Despiteitsadvantagetoexploreadditionalnetworksemantics,thetightdependencies 14

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betweeniterationsinevolutionarymodelsbringthebiggestobstacleintheprobabilisticanalysis[ 15 33 ].Structuralmodels,ontheotherhand,startwithagivenskeweddegreedistribution(e.g.,apower-lawdistributionbasedonthedegreesequencesofareal-worldnetwork[ 18 ])andgenerateagraphwiththedegreesequence,satisfyingcertainrandomnessproperties[ 2 37 82 ].Thegreatestadvantageofsuchstructuralmodelsistheirtractabilitytotheoreticalanalysis,duetoitsdiscardofdependenciesinevolutionarymodelsbytakingskeweddegreesequences[ 2 21 66 ].Althoughthetermcongurationisused,alotofmathematiciansalsonotedthisadvantagebyexploitingseveralpropertiesinstructuralrandomgraphmodels[ 14 68 69 ].Therefore,inthisdissertation,weusethewell-acceptedstructuralPLRGmodelin[ 2 ]inordertoexplorethepower-lawnetworksfromanin-depththeoreticalperspective.Giventheparametersand,thePLRGmodelisproposedasanstructuralapproachtoconstructa(,)power-lawgraphaccordingtoitsdegreesequence~d,whichconsistsofasequenceofintegers(1,...,1,2,...,2,...,)wherethenumberofiisequaltoyidenedintheaboveDenition 1 Denition2(Power-LawRandomGraph(PLRG)Model). Given~d=(d1,d2,...,dn)beasequenceofintegers(1,...,1,2,...,2,...,)wherethenumberofiisequaltoyi,thePLRGmodelgeneratesarandomgraphasfollows.ConsiderD=Pni=1dimini-nodeslyinginnclustersofeachsizediwhere1in,weconstructarandomperfectmatchingamongthemini-nodesandgenerateagraphonthenoriginalnodesassuggestedbythisperfectmatchinginthenaturalway:twooriginalnodesareconnectedbyanedgeifandonlyifatleastoneedgeintherandomperfectmatchingconnectsthemini-nodesoftheircorrespondingclusters. 1.2OptimizationProblemsinPower-LawGraphsAgreatnumberoflarge-scalenetworksinreallifearediscoveredtofollowapower-lawdistributionintheirdegreesequences,rangingfromtheInternet[ 34 ],theWorld-WideWeb(WWW)[ 4 ]tosocialnetworks[ 74 ].Thatis,thenumberofvertices 15

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withdegreeiisproportionaltoi)]TJ /F12 7.97 Tf 6.58 0 Td[(forsomeconstantinthesegraphs,whichiscalledpower-lawgraphs.Theobservationsshowthattheexponentialfactorrangesbetween1and4formostreal-worldnetworks[ 18 ].Intuitively,thefollowingtheoreticalquestionisraised:Whatarethedifferencesintermsofcomplexityhardnessandinapproximabilityfactorofseveraloptimizationproblemsbetweeningeneralgraphsandinpower-lawgraphs?Manyexperimentalresultsonrandompower-lawgraphsgiveusabeliefthattheproblemsmightbemucheasiertosolveonpower-lawgraphs.Eubanketal.[ 32 ]showedthatasimplegreedyalgorithmleadstoa1+o(1)approximationfactoronMINIMUMDOMINATINGSET(MDS)andMINIMUMVERTEXCOVER(MVC)onpower-lawgraphs(withoutanyformalproof)althoughMDSandMVChasbeenprovedNP-hardtobeapproximatedwithin(1)]TJ /F4 11.955 Tf 13.18 0 Td[()lognand1.366ongeneralgraphsrespectively[ 28 ].In[ 73 ],Gopalalsoclaimedthatthereexistsapolynomialtimealgorithmthatguaranteesa1+o(1)approximationoftheMVCproblemwithprobabilityatleast1)]TJ /F7 11.955 Tf 12.19 0 Td[(o(1).Unfortunately,thereisnosuchformalproofforthisclaimeither.Furthermore,severalpapersalsohavesometheoreticalguaranteesforsomeproblemsonpower-lawgraphs.Gkantsidisetal.[ 36 ]provedtheowthrougheachlinkisatmostO(nlog2n)onpower-lawrandomgraphswheretheroutingofO(dudv)unitsofowbetweeneachpairofverticesuandvwithdegreesduanddv.In[ 36 ],theauthorstakeadvantageofthepropertyofpower-lawdistributionbyusingthestructuralrandommodel[ 2 ]andshowthetheoreticalupperboundwithhighprobability1)]TJ /F7 11.955 Tf 12.31 0 Td[(o(1)andthecorrespondingexperimentalresults.Likewise,Jansonetal.[ 48 ]gaveanalgorithmthatapproximatedMAXIMUMCLIQUEwithin1)]TJ /F7 11.955 Tf 12.81 0 Td[(o(1)onpower-lawgraphswithhighprobabilityontherandompoissonmodelG(n,)(i.e.thenumberofverticeswithdegreeatleastidecreasesroughlyasn)]TJ /F8 7.97 Tf 6.58 0 Td[(i).Althoughtheseresultswerebasedonexperimentsandvariousrandommodels,theyraiseaninterestininvestigatinghardnessandinapproximabilityofoptimizationproblemsonpower-lawgraphs. 16

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Recently,Ferranteetal.[ 35 ]hadaninitialattemptonpower-lawgraphstoshowtheNP-hardnessofMAXIMUMCLIQUE(CLIQUE)andMINIMUMGRAPHCOLORING(COLORING)(>1)byconstructingabipartitegraphtoembedageneralgraphintoapower-lawgraphandNP-hardnessofMVC,MDSandMAXIMUMINDEPENDENTSET(MIS)(>0)basedontheiroptimalsubstructureproperties. 1.3VulnerabilityAssessmentofPower-LawNetworksMoststudiesinvestigatingthispower-lawpropertyhavebeenfocusedonhowsuchdegreeheterogeneitynaturecanimpacttherobustnessofnetworks[ 3 5 43 ],orhowonecanquicklyandefcientlygenerateanidealpower-lawnetworkwithagivendegreesequence[ 2 14 ].Focusingonthesecurityfactor,theworks[ 5 23 43 72 ]haveempiricallyshownthatpower-lawnetworksappearrobustunderrandomattacksandvulnerabletointentionalattacksviaexperimentalobservations.Nevertheless,thereareseveralimportantsecurityaspectsofthispropertythatareleftuntouched.Forinstance,arepower-lawnetworkssurelymorevulnerabletointentionalattacksthanrandomfailures?Howcanweaccuratelyassesstherobustnessofpower-lawnetworksundervariouskindsofthreat,e.g.,randomfailureandadversarialattack?Canwedesignmorestableandrobustpower-lawnetworksbyadjustingtheparameter?Anotherlimitationofthesepriorworksistheirheavydependenceontheexperimentsandfailurestooptimizethepower-lawnetworks.Inotherwords,wecannotapplythemtoenhancetherobustnessofpower-lawnetworks,andinthemeanwhilereducetheircosts.Toourbestknowledge,thisworkistherstattemptfromatheoreticalpointofviewtargetinginthetwoobjectivesmentionedabove:(1)assessingtheimpactofrandomandintentionalattacksonpower-lawnetworks;(2)optimizingpower-lawnetworksbasedontheirtolerationonthreatsandmaintenancecosts,whichareusedtoguaranteethenetworkfunctionalityandreliability. 17

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1.4OptimizationofPower-LawNetworksAlthoughpower-lawnetworksaremorerobustwhenissmaller,amajorityofreal-worldnetworksusuallyhavetheirexponentialfactorrangingfrom2to2.5ratherthansomesmallapproaching1orevenless.Thequestionsareintuitivelyraised:Isitbetterifreal-worldnetworksaredensersuchthattheycanbemorerobust?Whatcausesthemtobesparserthanourexpectation?Doesthereexistsomepotentialoptimizationfactors?Ontheotherhand,inordertooptimallymaintainthepower-lawnetworks,itisofgreatimportancetoassessthenetworkvulnerability,thatis,tostudyhowmuchthenetworkperformancereducesinvariouscasesofundesireddisruptions,suchasnaturaldisasters,unexpectedelementsfailures,orespeciallyadversarialattacks.Inatypicalattackingpointofview,anattackerwouldrstexploitthenetworkweaknesses,andthenonlyneedstotargetonsomecriticallinksornodeswhosecorruptionsbringthewholenetworkdowntoitsknees.Forinstance,anadversarialattacktoanyessentialInternetproviders,e.g.,tier-1ISPssuchasQwest,AT&TorSprintservers,oncesuccessful,maycausetremendousbreakdownstomillionsofcompanies'websitesandonlineservices.Inanaturaldisaster,anunexpectedearthquakemaydestroysomeimportantpowerlines,andconsequentlyleadtoalarge-areablackout.Therefore,itiscrucialtoexplorethenetworkvulnerability,i.e.,identifythosecruciallinksandnodes,beforehand. 1.5OutlineofDissertationTherestofdissertation,focusingonaddressingtheabovethreetopics,isorganizedasfollows:Chapter 2 presentsthehardnessandinapproximabilityresultsofclassicoptimizationprobleminpower-lawnetworks,inwhichweproposetwonoveltechniquestoembedad-boundedgraphintogeneralpower-lawgraphsandsimplepower-lawgraphsrespectively.Inaddition,wedesignaLow-DegreePercolation(LDP)AlgorithmFrameworkfortheseoptimizationproblems,andfurtherprovideatheoreticalframeworktoanalyzeapproximationratiosinpower-lawgraphs.InChapter 3 ,weexplorethe 18

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vulnerabilityofpower-lawnetworksviain-depthprobabilisticanalysis,underrandomfailures,intentionalattacks,andrandomcascadingfailures.Chapter 4 investigatestheoptimizationofpower-lawnetworks.Fromadesignperspective,weshowthat,inbothcommunicationandsocialcontext,thepower-lawnetworkswithexponentialfactorbetween1.8and2.5resultsintheoptimaldesign.Furthermore,inordertobetterprotectthepower-lawnetworks,westudyCLD,CNDandCCNDproblemstodetectcriticalelements.AftershowingtheNP-hardnessoftheseproblems,wedevelopHILPRandTRGAalgorithmstosolvetheminatimelymanner.ThewholedissertationisconcludedinChapter 5 19

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CHAPTER2HARDNESSANDAPPROXIMATIONALGORITHMSInthischapter,wedeveloptwonewtechniquesonoptimalsubstructureproblems,Cycle-BasedEmbeddingTechniqueandGraphicEmbeddingTechnique,toembedad-boundedgraphintoageneralpower-lawgraphandasimplepower-lawgraphrespectively.ThenweusethesetwotechniquestofurtherprovetheAPX-hardnessandtheinapproximabilityofMIS,MDS,andMVCongeneralpower-lawgraphsandsimplepower-lawgraphs.Theseinapproximabilityresultsonpower-lawgraphsareshowninTable 2-1 .Furthermore,theinapproximabilityresultsinCLIQUEandCOLORINGareshownbytakingadvantageofthereductionin[ 35 ].WealsoanalyzetherelationshipbetweenandconstantgreedyapproximationalgorithmsforMISandMDS.Inaddition,duetoalotofrecentstudiesinonlinesocialnetworksontheinuencepropagationproblem[ 54 55 ],weformulatethisproblemas-MinimumDominatingSet(-MDS)andshowithardtobeapproximatedwithin2)]TJ /F5 11.955 Tf 12.84 0 Td[((2+od(1))loglogd=logdfactorond-boundedgraphsunderuniquegamesconjecture,whichfurtherleadstothefollowinginapproximabilityresultonpower-lawgraphs(showninTable 2-1 ). Table2-1. InapproximabilityFactorsonPower-LawGraphswithExponentialFactor>1 Problem GeneralPower-LawGraph SimplePower-LawGraph MIS 1+1 140(2()3)]TJ /F9 7.97 Tf 6.59 0 Td[(1))]TJ /F4 11.955 Tf 11.96 0 Td[(" 1+1 1120()3)]TJ /F4 11.955 Tf 11.96 0 Td[(" MDS 1+1 390(2()3)]TJ /F9 7.97 Tf 6.59 0 Td[(1) 1+1 3120()3 MVC,-MDS 1+2(1)]TJ /F9 7.97 Tf 6.58 0 Td[((2+oc(1))loglogc logc) ()c+c1 (c+1) 1+2)]TJ /F9 7.97 Tf 6.58 0 Td[((2+oc(1))loglogc logc 2()c(c+1) CLIQUE O)]TJ /F7 11.955 Tf 5.48 -9.68 Td[(n1=(+1))]TJ /F12 7.97 Tf 6.59 0 Td[( COLORING O)]TJ /F7 11.955 Tf 5.48 -9.68 Td[(n1=(+1))]TJ /F12 7.97 Tf 6.59 0 Td[( a Conditions:MISandMDS:P6=NP;MVC,-MDS:uniquegamesconjecture;CLIQUE,COLORING:NP6=ZPP. b cisaconstantwhichisthesmallestdsatisfyingtheconditionin[ 10 ]. 2.1PreliminariesInthissection,werstrecallthedenitionofseveralclassicaloptimizationproblemsandformulatethenewoptimizationproblem-MinimumDominatingSet.Thenthe 20

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power-lawmodelandsomecorrespondingconceptsareproposed.Atlast,weintroducesomespecialgraphswhichwillbeusedintheanalysisthroughoutthewholepaper. 2.1.1ProblemDenitions Denition3(MAXIMUMINDEPENDENTSET). GivenanundirectedgraphG=(V,E),ndasubsetSVwiththemaximumsizesuchthatnotwoverticesinSareadjacent. Denition4(MINIMUMVERTEXCOVER). GivenanundirectedgraphG=(V,E),ndasubsetSVwiththeminimumsizesuchthatforeachedgeEatleastoneendpointbelongstoS. Denition5(MINIMUMDOMINATINGSET). GivenanundirectedgraphG=(V,E),ndasubsetSVwiththeminimumsizesuchthatforeachvertexvi2VnS,atleastoneneighborofvibelongstoS. Denition6(MAXIMUMCLIQUE). GivenanundirectedgraphG=(V,E),ndacliquewithmaximumsizewhereasubgraphofGiscalledacliqueifallitsverticesarepairwiseadjacent. Denition7(MINIMUMGRAPHCOLORING). GivenanundirectedgraphG=(V,E),labeltheverticesinVwithminimumnumberofcolorssuchthatnotwoadjacentverticessharethesamecolor.The-MinimumDominatingSetisdenedasgeneralversionofMDSproblem.Inthecontextofinuencepropagation,the-MDSproblemaimstondasubsetofnodeswithminimumsizesuchthatallnodesinthewholenetworkcanbeinuencedwithintrounds.Inparticular,anodeisinuencedwhenfractionofitsneighborsareinuenced.Forsimplicity,wedene-MDSprobleminthecasethatt=1. Denition8(-MINIMUMDOMINATINGSET). GivenanundirectedgraphG=(V,E),ndasubsetSVwiththeminimumsizesuchthatforeachvertexvi2VnS,jS\N(vi)jjN(vi)j. 21

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2.1.2SomeNotationsAgreatnumberofmodels[ 2 2 11 13 71 ]onpower-lawgraphsareemerginginthepastrecentyears.Inthischapter,wedotheanalysisbasedonthegeneral(,)model,thatis,thegraphsonlyconstrainedbythepower-lawdistributionindegreesequences.Werstdenethefollowingtwotypesofdegreesequences. Denition9(y-DegreeSequence). GivenagraphG=(V,E),they-degreesequenceofGisasequenceY=hy1,y2,...,yiwhereisthemaximumdegreeofGandyi=jfuju2V^deg(u)=igj. Denition10(d-DegreeSequence). GivenagraphG=(V,E),thed-degreesequenceofGisasequenceD=hd1,d2,...,dniofvertexinnon-increasingorderoftheirdegrees.Notethaty-degreesequenceandd-degreesequenceareinterchangeable.Givenay-degreesequenceY=hy1,y2,...,yi,thecorrespondingd-degreesequenceisD=h,,...,)]TJ /F5 11.955 Tf 12.95 0 Td[(1,)]TJ /F5 11.955 Tf 12.96 0 Td[(1,...,)]TJ /F5 11.955 Tf 12.95 0 Td[(1,...,1,...,1iwherethenumberiappearsyitimes.Becauseoftheirequivalence,wemayuseonlyy-degreesequenceord-degreesequenceorbothwithoutchangingthemeaningorvalidityofresults. Denition11(ContinuousSequence). Anintegersequencehd1,d2,...,dni,whered1d2dn,iscontinuousif81in)]TJ /F5 11.955 Tf 11.96 0 Td[(1,jdi)]TJ /F7 11.955 Tf 11.96 0 Td[(di+1j1. Denition12(GraphicSequence). AsequenceDissaidtobegraphicifthereexistsagraphsuchthatDisitsd-degreesequence. Denition13(DegreeSet). GivenagraphG,letDi(G)bethesetofverticesofdegreeionG.Furthermore,wedenethed-boundedgraphas Denition14(d-BoundedGraph). GivenagraphG=(V,E),Gisad-boundedgraphifthedegreeofanyvertexisupperboundedbyanintegerconstantd. 2.1.3SpecialGraphs Denition15(~d-RegularCycleRC~dn). Givenavector~d=(d1,...,dn),a~d-regularcycleRCdniscomposedoftwocycles.Eachcyclehasnverticesandtwoithverticesin 22

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ARC38 B~2)]TJ /F23 9.963 Tf 9.96 0 Td[(C28Figure2-1. SpecialGraphExamples:Theleftoneisa(3,3,3,3,3,3,3,3)-regularcycleandtherightoneisa(3,3,3,3)-branch-(2,2,2,2,2,2)-cycle.ThegreyverticesconsistoftheoptimalsolutionofMDSonthesetwospecialgraphs. eachcycleareadjacentwitheachotherbydi)]TJ /F5 11.955 Tf 12.36 0 Td[(2multi-edges.Thatis,~d-regularcycleRCdnhas2nverticesandthetwoithvertexhasthesamedegreedi.AnexampleRCd8isshowninFigure 2-1A Denition16(~-Branch-~d-Cycle~-BC~dn). Giventwovectors~d=(d1,...,dn)and~=(1,...,m),the~-branch-~d-cycleiscomposedofacyclewithanumberofverticesnsuchthateachvertexhasdegreediaswellasj~j=2appendantbranches,wherejjisaevennumber.Notethatany~-branch-~d-cyclehasj~jevennumberofverticeswithodddegrees.AnexampleisshowninFigure 2-1B 2.1.4ExistingInapproximabilityResultsHerewelistsomeinapproximabilityresultsintheliteraturetouselaterinourproofs. (1) Ind-boundedgraphs,MVCishardtobeapproximatedinto2)]TJ /F5 11.955 Tf 9.3 0 Td[((2+od(1))loglogd=logdforeverysufcientlylargeintegerdunderuniquegamesconjecture[ 10 20 ]. (2) In3-boundedgraphs,MISandMDSisNP-hardtobeapproximatedinto140 139)]TJ /F4 11.955 Tf 12.07 0 Td[("forany">0and391 390respectively[ 8 ]. (3) Maximumcliqueandminimumcoloringproblemishardtobeapproximatedinton1)]TJ /F12 7.97 Tf 6.58 0 Td[(ongeneralgraphsunlessNP=ZPP[ 41 ]. 23

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2.2InapproximabilityOptimalSubstructureFrameworkinPower-LawGraphsInthissection,weintroduceaframeworktoderivetheapproximationhardnessofoptimalsubstructureproblemsonpower-lawgraphs.Agraphoptimizationproblemissaidtosatisfyoptimalsubstructureifitsoptimalsolutionistheunionoftheoptimalsolutionsoneachconnectedcomponent.Therefore,whenagraphGisembeddedintoapower-lawgraphG0,theoptimalsolutioninG0consistsofasubsetoftheoptimalsolutioninG.Accordingtothisimportantproperty,wepresenttheInapproximabilityOptimalSubstructureFrameworktoprovetheinapproximabilityfactorifthereexistsaEmbedded-Approximation-PreservingReductionthatrelatestheapproximationhardnessingeneralgraphsandpower-lawgraphsbyguaranteeingtherelationshipbetweenthesolutionsintheoriginalgraphandtheconstructedgraph. Denition17(Embedded-Approximation-PreservingReduction). GivenanoptimalsubstructureproblemO,areductionfromaninstanceongraphG=(V,E)toanotherinstanceonapower-lawgraphG0=(V0,E0)iscalledembedded-approximation-preservingifitsatisesthefollowingproperties: (1) GisasubsetofmaximalconnectedcomponentsofG0; (2) TheoptimalsolutionofOonG0,OPT(G0),isupperboundedbyCOPT(G)whereCisaconstantcorrespondenttothegrowthoftheoptimalsolution. Theorem2.1(InapproximabilityOptimalSubstructureFramework). GivenanoptimalsubstructureproblemO,ifthereexistsanembedded-approximation-preservingreduc-tionfromagraphGtoanothergraphG0,wecanextracttheinapproximabilityfactorofOonG0using-inapproximabilityofOonG,whereislowerboundedbyC (C)]TJ /F9 7.97 Tf 6.59 0 Td[(1)+1and+C)]TJ /F9 7.97 Tf 6.58 0 Td[(1 CwhenOisamaximumandminimumoptimizationproblemrespectively. Proof. SupposethatthereexistsanalgorithmprovidingasolutionofOonG0withsizeatmosttimestheoptimalsolution.DenoteAandBtobethesizesoftheproducedsolutiononGandG0nGandAandBtobetheircorrespondingoptimalvalues.Hence,wehaveB(C)]TJ /F5 11.955 Tf 12.1 0 Td[(1)A.WiththecompletenessthatOPT(G)=A)OPT(G0)=B, 24

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thesoundnessleadstothelowerboundofwhichisdependentonthetypeofO,maximizationorminimizationproblem,asfollows.Case1:WhenOisamaximizationproblem,westartfromthedenitionofsoundnessasA+B(A+B) (2),AA+()]TJ /F5 11.955 Tf 11.95 0 Td[(1)B (2),AA+()]TJ /F5 11.955 Tf 11.95 0 Td[(1)(C)]TJ /F5 11.955 Tf 11.96 0 Td[(1)A (2)where( 2 )holdssinceBBand( 2 )holdssinceB(C)]TJ /F5 11.955 Tf 11.96 0 Td[(1)A.Ontheotherhand,itishardtoapproximateOwithinonG,thusA>A.Replaceittotheaboveinequality,wehave:AC (C)]TJ /F5 11.955 Tf 11.96 0 Td[(1)+1Case2:WhenOisaminimizationproblem,sinceBB,similarlyA+B(A+B),AA+()]TJ /F5 11.955 Tf 11.96 0 Td[(1)B,AA+()]TJ /F5 11.955 Tf 11.96 0 Td[(1)(C)]TJ /F5 11.955 Tf 11.95 0 Td[(1)AThenfromA>A,<+()]TJ /F5 11.955 Tf 11.96 0 Td[(1)(C)]TJ /F5 11.955 Tf 11.95 0 Td[(1),>+C)]TJ /F5 11.955 Tf 11.96 0 Td[(1 C 2.3HardnessandInapproximabilityofOptimalSubstructureProblems 2.3.1GeneralCycle-BasedEmbeddingTechniqueInthissection,weproposeaGeneralCycle-BasedEmbeddingTechniqueon(,)power-lawgraphswith>1.Thebasicideaistoembedanarbitraryd-boundedgraph 25

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intopower-lawgraphsusinga~d1-regularcycle,a~-branch-~d2-cycleandanumberofcliquesK2,where~d1,~d2and~aredenedbyand.Beforediscussingthemainembeddingtechnique,werstshowthatmostoptimalsubstructureproblemscanbepolynomiallysolvedinboth~d-regularcyclesand~-branch-~d-cycle.Inthiscontext,thecycle-basedembeddingtechniquehelpstoprovethecomplexityoftheseoptimalsubstructureproblemsonpower-lawgraphsaccordingtotheircorrespondingcomplexityresultsongeneralboundedgraphs. Lemma1. MDS,MVCandMISarepolynomiallysolvableon~d-regularcycles. Proof. HerewejustproveMDSproblemispolynomiallysolvableon~d-regularcycles.Thealgorithmissimple.Fromanarbitrarilyvertex,weselectthevertexontheothercycleintwohops.Thealgorithmwillterminateuntilallverticesaredominated.Nowwewillshowthatthisgivestheoptimalsolution.LettakeRC38asanexample.AsshowninFigure 2-1A ,thesizeofMDSis4.Noticethateachvertexcandominateexact3vertices,thatis,4verticescandominateexactly12vertices.However,inRC38,therearealtogether16vertices,whichhavetobedominatedbyatleast4verticesapartfromtheverticesinMDS.Thatis,thealgorithmreturnsanoptimalsolution.TheproofofMVCandMISissimilar. Lemma2. MDS,MVCandMISispolynomiallysolvableon~-branch-~d-cycles. Proof. AgainweshowtheproofofMDS.Firstweselecttheverticesconnectingboththebranchesandthecycle.Thenbyremovingthebranches,wewillhavealinegraphregardlessofself-loops,onwhichMDSispolynomiallysolvable.ItiseasytoseethatthesizeofMDSwillincreaseifanyonevertexconnectingboththebranchandthecycleinMDSisreplacedbysomeothervertices.TheproofofMISissimilar.NotethattheoptimalsolutionforMVCconsistsofallverticessincealledgesneedtobecovered. Theorem2.2(Cycle-BasedEmbeddingTechnique). Anyd-boundedgraphGdcanbeembeddedintoapower-lawgraphG(,)with>1suchthatGdisamaximal 26

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componentandmostoptimalsubstructureproblemscanbepolynomiallysolvableonG(,)nGd. Proof. Withthegiven,wechoosetobemaxflnmax1idfniig,lndg.Basedon(i)=be=ic)]TJ /F7 11.955 Tf 20.69 0 Td[(niwhereni=0wheni>d,weconstructthepower-lawgraphG(,)asthefollowingAlgorithm 1 .Thelaststepholdssincethenumberofverticesofodddegreeshastobeeven.FromStep1,weknowe=maxfmax1idfniig,dgdn,thatis,thenumberofverticesNingraphG(,)satisesN()dn,whichmeansthatN=nisaconstant.AccordingtoLemma 1 andLemma 2 ,sinceG(,)nGdiscomposedofa~d1-regularcycleanda~d12-branch-~d2-cycle,itcanbepolynomiallysolvable.NotethatthenumberofverticesinLisatmostsincethereisatmostoneleftoververtexofeachdegree. Algorithm1:CycleEmbeddingAlgorithm 1 maxflnmax1idfniig,lndg; 2For(1)verticesofdegree1,addb(1)=2cnumberofcliquesK2; 3For(2)verticesofdegree2,addacyclewiththesize(2); 4Forallverticesofdegreelargerthan2andsmallerthan,constructa~d1-regularcyclewhere~d1isavectorcomposedofb(i)=2cnumberofelementsiforallisatisfying(i)>0; 5ForallleftoverisolatedverticesLsuchthat(i))]TJ /F5 11.955 Tf 11.95 0 Td[(2b(i)=2c=1,constructa~d12-branch-~d22-cycle,where~d12and~d22arethevectorscontainingoddandevenelementscorrespondenttotheverticesofoddandevendegreesinLrespectively. 2.3.2APX-HardnessInthissection,weprovethatMIS,MDS,MVCremainAPX-hardevenonpower-lawgraphs. Theorem2.3. MDSisAPX-hardonpower-lawgraphs. 27

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Proof. AccordingtoTheorem 2.2 ,weusethecycle-basedembeddingtechniquetoshowL-reductionfromMDSonanyd-boundedgraphGdtoMDSonapower-lawgraphG(,)sinceMDSisprovenAPX-hardond-boundedgraphs[ 51 ].LettingbeafeasiblesolutiononGd,wecanconstructMDSinG0suchthatMDSonaK2is1,n=4ona~d-regularcycleandn=3onacycleanda~-branch-~d-cycle.Therefore,forasolutiononGd,wehaveasolution'onG(,)tobe'=+n1=2+n2=3+n3=4,wheren1,n2andn3correspondsto(1),(2)[Landallleftoververtices.Hence,wehaveOPT(')=OPT()+n1=2+n2=3+n3=4.Ononehand,forad-boundedgraphwithverticesn,theoptimalMDSislowerboundedbyn=(d+1).Thus,weknowOPT(')=OPT()+n1=2+n2=3+n3=4OPT()+(N)]TJ /F7 11.955 Tf 11.95 0 Td[(n)=2OPT()+(()d)]TJ /F5 11.955 Tf 11.95 0 Td[(1)n=2OPT()+(()d)]TJ /F5 11.955 Tf 11.96 0 Td[(1)(d+1)OPT()=2=1+(()d)]TJ /F5 11.955 Tf 11.96 0 Td[(1)(d+1)=2OPT()whereNisthenumberofverticesinG(,).Ontheotherhand,withjOPT())]TJ /F4 11.955 Tf 12.16 0 Td[(j=jOPT('))]TJ /F4 11.955 Tf 12.16 0 Td[('j,weprovedtheL-reductionwithc1=1+(()d)]TJ /F5 11.955 Tf 11.96 0 Td[(1)(d+1)=2andc2=1. Theorem2.4. MVCisAPX-hardonpower-lawgraphs. Proof. Inthisproof,weshowL-reductionfromMVCond-boundedgraphGdtoMVConpower-lawgraphG(,)usingcycle-basedembeddingtechnique.LetbeafeasiblesolutiononGd.Weconstructthesolution'+(N)]TJ /F7 11.955 Tf -415.86 -23.9 Td[(n)sincetheoptimalsolutionofMVCisn=2onK2,cycle,~d-regularcycleandnon~-branch-~d-cycle.Therefore,sincetheoptimalMVConad-boundedgraphislowerboundedbyn=(d+1),wehaveOPT(')1+(()d)]TJ /F5 11.955 Tf 11.95 0 Td[(1)(d+1)OPT() 28

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Ontheotherhand,withjOPT())]TJ /F4 11.955 Tf 12.16 0 Td[(j=jOPT('))]TJ /F4 11.955 Tf 12.16 0 Td[('j,weprovedtheL-reductionwithc1=1+(()d)]TJ /F5 11.955 Tf 11.96 0 Td[(1)(d+1)andc2=1. Corollary1. MISisAPX-hardonpower-lawgraphs. 2.3.3InapproximabilityFactorsInthissection,weshowtheinapproximabilityfactorsonMIS,MVCandMDSonpower-lawgraphsrespectivelyusingtheresultsinsection 2.1.4 Theorem2.5. Forany">0,thereisno1+1 140(2()3)]TJ /F9 7.97 Tf 6.58 0 Td[(1))]TJ /F4 11.955 Tf 11.97 .01 Td[("approximationalgorithmforMaximumIndependentSetonpower-lawgraphs. Proof. Inthisproof,weconstructthepower-lawgraphG(,)basedoncycle-basedembeddingtechniqueinTheorem 2.2 fromd-boundedgraphGd.Letand'befeasiblesolutionsofMISonGdandG(,).ThenOPT(')composedofOPT(),cliqueK2,cycle,~d-regularcycleand~-branch-~d-cyclesareallexactlyhalfnumberofvertices.Hence,wehaveOPT(')=OPT()+(N)]TJ /F7 11.955 Tf 12.8 0 Td[(n)=2wherenandNisthenumberofverticesinGdandG(,)respectively.SinceOPT()n=(d+1)ond-boundedgraphsforMISandN()dn,wefurtherhaveC=1+(()d)]TJ /F9 7.97 Tf 6.58 0 Td[(1)(d+1) 2fromOPT(')=OPT()+N)]TJ /F7 11.955 Tf 11.95 0 Td[(n 2OPT()+(()d)]TJ /F5 11.955 Tf 11.96 0 Td[(1) 2nOPT()+(()d)]TJ /F5 11.955 Tf 11.96 0 Td[(1)(d+1) 2OPT()=1+(()d)]TJ /F5 11.955 Tf 11.95 0 Td[(1)(d+1) 2OPT()Accordingto=140 139)]TJ /F4 11.955 Tf 13.58 0 Td[("0forany0>0on3-boundedgraphs,thentheinapproximabilityfactorcanbederivedfrominapproximabilityoptimalsubstructureframeworkas>C (C)]TJ /F5 11.955 Tf 11.96 0 Td[(1)+1>1+1 140C)]TJ /F4 11.955 Tf 11.96 0 Td[("=1+1 140(2()3)]TJ /F5 11.955 Tf 11.95 0 Td[(1))]TJ /F4 11.955 Tf 11.96 0 Td[(" 29

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wherethelaststepfollowsfromd=3. Theorem2.6. Thereisno1+1 390(2()3)]TJ /F9 7.97 Tf 6.59 0 Td[(1)approximationalgorithmforMinimumDominatingSetonpower-lawgraphs. Proof. Inthisproof,weconstructthepower-lawgraphG(,)basedoncycle-basedembeddingtechniqueinTheorem 2.2 fromd-boundedgraphGd.Letand'befeasiblesolutionsofMDSonGdandG(,).TheoptimalMDSonOPT(),cliqueK2,cycle,~d-regularcycleand~-branch-~d-cyclesaren=2,n=4andn=3respectively.Letand'befeasiblesolutionsofMDSonGdandG(,).ThenwehaveC=1+(()d)]TJ /F9 7.97 Tf 6.58 0 Td[(1)(d+1) 2similarastheproofinTheorem 2.5 .Accordingto=391 390in3-boundedgraphs,thentheinapproximabilityfactorcanbederivedfrominapproximabilityoptimalsubstructureframeworkas>1+)]TJ /F5 11.955 Tf 11.95 0 Td[(1 C=1+1 390(2()3)]TJ /F5 11.955 Tf 11.95 0 Td[(1)wherethelaststepfollowsfromd=3. Theorem2.7. MVCishardtobeapproximatedwithin1+2(1)]TJ /F9 7.97 Tf 6.58 0 Td[((2+oc(1))loglogc logc) ()c+c1 (c+1)onpower-lawgraphsunderuniquegamesconjecture. Proof. Byconstructingthepower-lawgraphG(,)basedoncycle-basedembeddingtechniqueinTheorem 2.2 fromd-boundedgraphGd,TheoptimalMVConcliqueK2,cycle,~d-regularcyclearehalfnumberofverticeswhiletheoptimalMVCon~-branch-~d-cyclesareallvertices.Thus,wehaveC=1+()d)]TJ /F9 7.97 Tf 6.59 0 Td[(1+d1 (d+1) 2since 30

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OPT(')OPT()+N)]TJ /F7 11.955 Tf 11.96 0 Td[(n)]TJ /F5 11.955 Tf 11.96 0 Td[( 2+OPT()+(()d)]TJ /F5 11.955 Tf 11.96 0 Td[(1)n+n1 d 2 (2) =OPT()+()d)]TJ /F5 11.955 Tf 11.95 0 Td[(1+d n1)]TJ /F16 5.978 Tf 7.53 2.35 Td[(1 n 2 (2) OPT()+()d)]TJ /F5 11.955 Tf 11.95 0 Td[(1+d (d+1)1)]TJ /F16 5.978 Tf 7.53 2.35 Td[(1 (d+1) 2OPT() (2) 0@1+()d)]TJ /F5 11.955 Tf 11.95 0 Td[(1+d1 (d+1) 21AOPT() (2) whereand'befeasiblesolutionsofMVConGdandG(,),isthemaximumdegreeinG(,).Theinequality( 2 )holdssincethereareatmostverticesin~-branch-~d-cycle,i.e.=e=n1=d;( 2 )holdssincethereareatleastd+1verticesinad-boundedgraphandtheoptimalMVCinad-boundedgraphisatleastn=(d+1).Accordingto=2)]TJ /F5 11.955 Tf 11.99 0 Td[((2+od(1))loglogd=logd,thentheinapproximabilityfactorcanbederivedfrominapproximabilityoptimalsubstructureframeworkas>1+)]TJ /F5 11.955 Tf 11.96 0 Td[(1 C1+21)]TJ /F5 11.955 Tf 11.95 0 Td[((2+oc(1))loglogc logc ()c+c1 (c+1)wherecisthesmallestdsatisfyingtheconditionin[ 10 ].Thelastinequalityholdssincefunctionf(x)=(1)]TJ /F5 11.955 Tf 12.01 0 Td[((2+ox(1))loglogx=logx)=g(x)(x+1)ismonotonouslydecreasingwhenf(x)>0forallx>0wheng(x)ismonotonouslyincreasing. Theorem2.8. -PDSishardtobeapproximatedinto2)]TJ /F5 11.955 Tf 10.86 0 Td[((2+od(1))loglogd logdond-boundedgraphsunderuniquegamesconjecture. Proof. Inthisproof,weshowthegap-preservingfromMVCon(d=)-boundedgraphG=(V,E)to-PDSond-boundedgraphG0=(V0,E0).w.l.o.g.,weassumethatdandd=areintegers.WeconstructagraphG0=(V0,E0)byaddingnewverticesand 31

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edgestoGasfollows.Foreachedge(vi,vj)2E,createknewverticesv1ij,...,vkijwhere1kb1=cand1=2.Thenweadd2knewedges(vlij,vi)and(vlij,vj)foralll2[1,k]asshowninFigure 2-2 .Clearly,G0=(V0,E0)isad-boundedgraph. AInstanceG=(V,E) BReducedInstanceG0=(V0,E0)Figure2-2. TheReductionfromMVCto-MDS Letand'befeasiblesolutionstoMVConGandG0respectively.WeclaimthatOPT()=OPT(').Ononehand,ifS=fv1,v2,...,vjg2VistheminimumvertexcoveronG.Thenfv1,v2,...,vjgisa-PDSonG0becauseeachvertexinVhasofallneighborsinMVCandeverynewvertexinV0nVhasatleastoneoftwoneighborsinMVC.ThusOPT()OPT(').Onetheotherhand,wecanprovethatOPT(')doesnotcontainnewvertices,thatis,V0nV.Consideravertexvi2V,ifvi2OPT('),thenewverticesvlijforallvj2N(vi)andalll2[1,k]arenotneededtobeselected.Ifvi62OPT('),ithastobedominatedbyproportionofitsallneighbors.Thatis,foreachedge(vi,vj)incidenttovi,eithervjorallvlijhavetobeselectedsinceeveryvlijhastobeeitherselectedordominated.IfallvlijareselectedinOPT(')forsomeedge(vi,vj),vjisstillnotdominatedbyenoughverticesiftherearesomemoreedgesincidenttovjandthenumberofverticesvlijkisgreatthan1,thatis,b1=c1.Inthiscase,vjwillbeselectedtodominateallvlij.Thus,OPT(')doesnotcontainnewvertices.SincetheverticesinVselectedisasolutionto-MDS,thatis,foreachvertexviingraphG,viwillbeselected 32

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oratleastthenumberofneighborsofviwillbeselected.Therefore,theverticesinOPT(')consistofavertexcoverinG.ThusOPT()OPT(').Thenweshowthecompletenessandsoundnessasfollows. IfOPT()=m)OPT(')=m IfOPT()>2)]TJ /F34 10.909 Tf 10.91 0 Td[((2+od(1))loglog(d=2) log(d=2)m)OPT(')>2)]TJ /F34 10.909 Tf 10.9 0 Td[((2+od(1))loglogd logdmOPT(')>2)]TJ /F5 11.955 Tf 11.96 0 Td[((2+od(1))loglog(d=) log(d=)m>2)]TJ /F5 11.955 Tf 11.95 0 Td[((2+od(1))loglogd logdmsincethefunctionf(x)=2)]TJ /F5 11.955 Tf 12.51 0 Td[(loglogx=logxismonotonouslyincreasingforanyx>0. Theorem2.9. -PDSishardtobeapproximatedinto1+2(1)]TJ /F9 7.97 Tf 6.58 0 Td[((2+oc(1))loglogc logc) 2+(()c)]TJ /F9 7.97 Tf 6.59 0 Td[(1)(c+1)onpower-lawgraphsunderuniquegamesconjecture. Proof. Byconstructingthepower-lawgraphG(,)basedoncycle-basedembeddingtechniqueinTheorem 2.2 fromd-boundedgraphGd,AccordingtotheoptimalMVConOPT(),cliqueK2,cycle,~d-regularcycleand~-branch-~d-cycles,wehaveC=1+(()d)]TJ /F9 7.97 Tf 6.58 0 Td[(1)(d+1) 2fromOPT(')=OPT()+n1=2+f()n2+g()n3OPT()+N)]TJ /F7 11.955 Tf 11.96 0 Td[(n 21+(()d)]TJ /F5 11.955 Tf 11.96 0 Td[(1)(d+1) 2OPT()wheref()=8>><>>:1 4,1 31 3,1 3<1 2,g()=1 3forall1 2and,'befeasiblesolutionsofMVConGdandG(,).n1,n2andn3arecorrespondenttothenumberofverticesincliquesK2,cycle,~d-regularcycleand~-branch-~d-cycle. 33

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Accordingto=2)]TJ /F5 11.955 Tf 11.99 0 Td[((2+od(1))loglogd=logd,thentheinapproximabilityfactorcanbederivedfrominapproximabilityoptimalsubstructureframeworkas>1+)]TJ /F5 11.955 Tf 11.96 0 Td[(1 C1+21)]TJ /F5 11.955 Tf 11.95 0 Td[((2+oc(1))loglogc logc 2+(()c)]TJ /F5 11.955 Tf 11.95 0 Td[(1)(c+1)wherecisthesmallestdsatisfyingtheconditionin[ 10 ].Thelastinequalityholdssincefunctionf(x)=(1)]TJ /F5 11.955 Tf 12.01 0 Td[((2+ox(1))loglogx=logx)=g(x)(x+1)ismonotonouslydecreasingwhenf(x)>0forallx>0wheng(x)ismonotonouslyincreasing. 2.4MoreInapproximabilityResultsonSimplePower-LawGraphs 2.4.1GeneralGraphicEmbeddingTechniqueInthissection,weintroduceageneralgraphicembeddingtechniquetoembedadboundedgraphintoasimplepower-lawgraph.Beforepresentingtheembeddingtechnique,werstshowthatagraphcanbeconstructedinpolynomialtimefromaclassofintegersequences. Lemma3. GivenasequenceofintegersD=hd1,d2,...,dniwhichisnon-increasing,continuousandthenumberofelementsisatleastastwiceasthelargestelementinD,i.e.n2d1,itispossibletoconstructasimplegraphGwhosed-degreesequenceisDinpolynomialtimeO(n2logn). Proof. StartingwithasetofindividualverticesSofdegree0andjSj=n,weiterativelyconnectverticestogethertoincreasetheirdegreesuptogivendegreesequence.Ineachstep,theleftoververtexofhighestdegreeisconnectedtootherverticesonebyoneinthedecreasingorderoftheirdegrees.ThenthesequenceDwillberesortedandallzeroelementswillberemoved.ThealgorithmstopsuntilDisempty.Thewholealgorithmisshownasfollows(Algorithm 2 ). Aftereachwhileloop,thenewdegreesequence,calledD0,isstillcontinuousanditsnumberofelementsisatleastastwiceasitsmaximumelement.Toshowthis,we 34

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Algorithm2:GraphicSequenceConstructionAlgorithm Input:d-degreesequenceD=hd1,d2,...,dniwhered1d2...dnOutput:GraphH 1whileD6=;do 2Connectvertexofd1toverticesofd2,d3,...,dd1+1; 3d1 0; 4fori=2tod1+1do 5di di)]TJ /F5 11.955 Tf 11.96 0 Td[(1; 6end 7SortDinnon-increasingorder; 8RemoveallzeroelementsinD; 9end considerthreecases:(1)IfthemaximumdegreeinD0remainsthesame,thereareatleastd1+2verticesinD.SinceDiscontinuous,thenumberofelementsinDisatleastd1+2+d1)]TJ /F5 11.955 Tf 12.69 0 Td[(1,thatis,2d1+1.Therefore,thenumberofelementsinD0is2d1,i.e.n2d1stillholds.(2)IfthemaximumdegreeinD0isdecreasedby1,thereareatleast2elementsofdegreed1inD.Thus,atmostoneelementinDwillbecome0.Thenwehaven2d1)]TJ /F5 11.955 Tf 12.12 0 Td[(2=2(d1)]TJ /F5 11.955 Tf 12.12 0 Td[(1).(3)IfthemaximumdegreeinD0isdecreasedby2,thereareatmosttwoelementinDbecoming0.Thus,n2d1)]TJ /F5 11.955 Tf 11.96 0 Td[(3>2(d1)]TJ /F5 11.955 Tf 11.96 0 Td[(2).ThetimecomplexityofthealgorithmisO(n2logn)sincethereareatmostniterationsandeachiterationtakesatmostO(nlogn)tosortthenewsequenceD. Theorem2.10(GraphicEmbeddingTechnique). Anyd-boundedgraphGdcanbeembeddedintoasimplepower-lawgraphG(,)with>1inpolynomialtimesuchthatGdisamaximalcomponentandthenumberofverticesinG(,)canbepolynomiallyboundedbythenumberofverticesinGd. Proof. Givenad-boundeddegreegraphGd=(V,E)and>1,weconstructapower-lawgraphG(,)ofexponentialfactorwhichincludesGdasasetofmaximalcomponents.TheconstructionisshownasAlgorithm 3 35

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Algorithm3:GraphicEmbeddingAlgorithm 1 maxf )]TJ /F9 7.97 Tf 6.58 0 Td[(1(ln4+lnd),ln2+lnn+lndgandcorrespondingG(,); 2Dbethed-degreesequenceofG(,)nGd; 3ConstructG(,)nGdusingAlgorithm 2 Accordingtothelemma 3 ,theaboveconstructionisvalidandnishesinpolynomialtime.ThenweshowthatNisupperboundedby()2dn,wherenandNarethenumberofverticesinGdandG,respectively.Fromtheconstruction,weknoweither )]TJ /F5 11.955 Tf 11.95 0 Td[(1(ln4+lnd))ln4+lnd+=)e d4e orln2+lnn+lnd)e d2nTherefore,e d2e +n.Notethate disthenumberofverticesofdegreed.Inaddition,Ghasatmostnverticesofdegreed,soDiscontinuousdegreesequenceandhasthenumberofverticesatleastastwiceasthemaximumdegree.Inaddition,whennislargeenough,wehave=ln2+lnn+lnd.Hence,thenumberofverticesNinG,isboundasN()e=2()dn,i.e.thenumberofverticesofG,ispolynomialboundedbythenumberofverticesinGd. 2.4.2InapproximabilityofMIS,MVCandMDS Theorem2.11. Forany">0,itisNP-hardtoapproximateMaximumIndependentSetwithin1+1 1120()3)]TJ /F4 11.955 Tf 11.95 0 Td[("onsimplepower-lawgraphs. Proof. Inthisproof,weconstructthesimplepower-lawgraphG(,)basedongraphicembeddingtechniqueinTheorem 2.10 fromd-boundedgraphGd.Letand'befeasiblesolutionsofMISonGdandG(,).SinceOPT()n=(d+1)ond-boundedgraphsandN2()dn,wefurtherhaveC=2()d(d+1)from 36

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OPT(')N2()dn2()d(d+1)OPT()Accordingto=140 139)]TJ /F4 11.955 Tf 13.58 0 Td[("0forany0>0on3-boundedgraphs,thentheinapproximabilityfactorcanbederivedfrominapproximabilityoptimalsubstructureframeworkas>C (C)]TJ /F5 11.955 Tf 11.95 0 Td[(1)+1=1+1 140C)]TJ /F5 11.955 Tf 11.96 0 Td[(1)]TJ /F4 11.955 Tf 11.95 0 Td[(">1+1 1120()3)]TJ /F4 11.955 Tf 11.95 0 Td[(" Theorem2.12. ItisNP-hardtoapproximateMinimumDominatingSetwithin1+1 3120()3onpower-lawgraphs. Proof. FromtheproofofTheorem 2.11 ,wehaveC=2()d(d+1).Thenaccordingto=391 390on3-boundedgraphs,wehave>1+)]TJ /F5 11.955 Tf 11.96 0 Td[(1 C1+1 3120()3 Theorem2.13. Thereisno1+2)]TJ /F9 7.97 Tf 6.59 0 Td[((2+oc(1))loglogc logc 2()c(c+1)approximationalgorithmofMinimumVertexCoveronpower-lawgraphsunderuniquegamesconjecture. Proof. SimilarastheproofofTheorem 2.12 ,wehaveC=2()d(d+1).Thenaccordingto=2)]TJ /F5 11.955 Tf 12.18 0 Td[((2+od(1))loglogd=logd,thentheinapproximabilityfactorcanbederivedfrominapproximabilityoptimalsubstructureframeworkas>1+)]TJ /F5 11.955 Tf 11.95 0 Td[(1 C1+2)]TJ /F5 11.955 Tf 11.96 0 Td[((2+oc(1))loglogc logc 2()c(c+1)wherecisthesmallestdsatisfyingtheconditionin[ 10 ]. 37

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Theorem2.14. Thereisno1+2)]TJ /F9 7.97 Tf 6.58 0 Td[((2+oc(1))loglogc logc 2()c(c+1)approximationalgorithmforMinimumPositiveDominatingSetonpower-lawgraphs. Proof. SimilarasTheorem 2.14 ,theprooffollowsfromTheorem 2.8 2.4.3MaximumClique,MinimumColoring Lemma4(Ferranteetal.[ 35 ]). LetG=(V,E)beasimplegraphwithnverticesand1.Letmaxf4,logn+log(n+1)g.Then,G2=GnG1isabipartitegraph. Lemma5. Givenafunctionf(x)(x2Z,f(x)2Z+)monotonouslydecreases,thenPxf(x)Rxf(x). Corollary2. ePe=i=1)]TJ /F9 7.97 Tf 7.03 -4.97 Td[(1 d<(e)]TJ /F7 11.955 Tf 11.96 0 Td[(e=)=()]TJ /F5 11.955 Tf 11.95 0 Td[(1). Theorem2.15. MaximumCliquecannotbeapproximatedwithinO)]TJ /F7 11.955 Tf 5.48 -9.69 Td[(n1=(+1))]TJ /F12 7.97 Tf 6.59 0 Td[(onlargepower-lawgraphswith>1andn>54forany>0unlessNP=ZPP. Proof. In[ 35 ],theauthorsprovedthehardnessofMaximumCliqueproblemonpower-lawgraphs.Hereweusethesameconstruction.AccordingtoLemma 27 ,G2=GnG1isabipartitegraphwhenmaxf4,logn+log(n+1)gforany1.LetbeasolutionongeneralgraphGand'beasolutiononpower-lawgraphG2.Weshowthecompletenessandsoundness. IfOPT()=m)OPT(')=mIfOPT()2ongraphG,wecansolvecliqueprobleminpolynomialtimebyiteratingtheedgesandtheirendpointsonebyone.However,Gisnotageneralgraphinthiscase.w.l.o.g,assumingOPT()>2,thenOPT(')=OPT()>2sincethemaximumcliqueonbipartitegraphis2. IfOPT()m=n1)]TJ /F12 7.97 Tf 6.59 0 Td[()OPT(')54.AccordingtoLemma 27 ,let=logn+log(n+1).FromCorollary 2 ,wehaveN=eXi=11 i
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Corollary3. MinimumColoringproblemcannotbeapproximatedwithinO)]TJ /F7 11.955 Tf 5.48 -9.69 Td[(n1=(+1))]TJ /F12 7.97 Tf 6.59 0 Td[(onlargepower-lawgraphswith>1andn>54forany>0unlessNP=ZPP. 2.5RelationshipbetweenandApproximationHardnessAsshowninprevioussections,manyhardnessandinapproximabilityresultsaredependenton.Inthissection,weanalyzethehardnessofsomeoptimalsubstructureproblemsbasedonbyshowingthattrivialgreedyalgorithmscanachieveconstantguaranteefactorsforMISandMDS. Lemma6. When>2,thesizeofMDSofapower-lawgraphisgreaterthanCnwherenisthenumberofvertices,Cissomeconstantonlydependenton. Proof. LetS=(v1,v2,...,vt)ofdegreesd1,d2,...,dtbetheMDSofpower-lawgraphG(,).Observingthatthetotaldegreesofverticesindominatingsetmustbeatleastthenumberofverticesoutsidethedominatingset,wehavePi=ti=1dijVnSj.Withagiventotaldegree,asetofverticeshasminimumsizewhenitincludestheverticesofhighestdegrees.Sincethefunction()]TJ /F5 11.955 Tf 12.2 0 Td[(1)=P1i=11 i)]TJ /F16 5.978 Tf 5.76 0 Td[(1convergeswhen>2,thereexistsaconstantt0=t0()suchthatXi=t0ie it0Xi=1e iwhereisanylargeenoughconstant.ThusthesizeofMDSisatleastXi=t0e i ())]TJ /F8 7.97 Tf 11.95 15.05 Td[(t0)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xi=11 i!eCjVjwhereC=(())]TJ /F6 11.955 Tf 11.96 8.96 Td[(Pt0i=11 i)=(()). Considerthegreedyalgorithmwhichselectsfromtheverticesofthehighestdegreetothelowest.Intheworstcase,itselectsallverticeswithdegreegreaterthan1andahalfofverticeswithdegree1toformadominatingset.Theapproximationfactorofthissimplealgorithmisaconstant. 39

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Corollary4. Givenapower-lawgraphwith>2,thegreedyalgorithmthatselectsverticesindecreasingorderofdegreesprovidesadominatingsetofsizeatmostPi=2e=i+1 2e(())]TJ /F5 11.955 Tf 11.77 0 Td[(1=2)e.Thustheapproximationratiois(())]TJ /F9 7.97 Tf 12.97 4.71 Td[(1 2)=(())]TJ /F6 11.955 Tf -458.7 -14.94 Td[(Pt0i=11=i).LetusconsideranothermaximizationproblemMIS,weproposeagreedyalgorithmPower-law-Greedy-MISasfollows.Wesorttheverticesinnon-increasingorderofdegreesandstartcheckingfromthevertexoflowestdegree.Ifthevertexisnotadjacenttoanyselectedvertex,itisselected.Thesetofselectedverticesformsanindependentsetwiththesizeatleastahalfthenumberofverticesofdegree1whichise=2.ThesizeofMISisatmostahalfofnumberofvertices.Thus,thefollowinglemmaholds. Lemma7. Power-law-Greedy-MIShasfactor1=(2())onpower-lawgraphswith>1. 2.6MinorNP-HardnessonSimplePower-LawGraphsfor<1Inthesection,weshowsomeminorNP-hardnessofoptimalsubstructureproblemsonsimplepower-lawgraphsforsmall<1. Denition18(EligibleSequences). AsequenceofintegersS=hs1,...,sniiseligibleifs1s2...snandfS(k)0forallk2[n],wherefS(k)=k(k)]TJ /F5 11.955 Tf 11.95 0 Td[(1)+nXi=k+1minfk,sig)]TJ /F8 7.97 Tf 26.81 14.94 Td[(kXi=1siErdosandGallai[ 31 ]showedthatanintegersequenceisgraphic-d-degreesequenceofangraph,ifandonlyifitiseligibleandthetotalofallelementsiseven.ThenHavelandHakimi[ 16 ]gaveanalgorithmtoconstructasimplegraphfromadegreesequence.Wenowprovethefollowingeligibleembeddingtechniquebasedonthisresult. Theorem2.16(EligibleEmbeddingTechnique). GivenanundirectedsimplegraphG=(V,E),0<<1,thereexistspolynomialtimealgorithmtoconstructapower-lawgraphG0=(V0,E0)ofexponentialfactorsuchthatGisasetofmaximalcomponentsofG0. 40

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Proof. ToconstructG0,wechoose=maxfln(n)]TJ /F5 11.955 Tf 12.89 0 Td[(1)+ln(n+2),3ln2g.Thenbe=((n)]TJ /F5 11.955 Tf 12.36 0 Td[(1))c>n+2,i.e.thereareatleast2verticesofdegreedinG0nGiftherearealeast2verticesofdegreedinG0.Accordingtothedenition,thetotaldegreesofallverticesinG0andGareeven.Therefore,thelemmawillfollowifweprovethatthedegreesequenceDofG0nGiseligible.InD,themaximumdegreeisbe=c.Thereisonlyonevertexofdegreeiif1e=i<2,i.e.e=i>(e=2)1=.LetusconsiderfD(k)intwocases:Case1:ke==2fD(k)=k(k)]TJ /F5 11.955 Tf 11.95 0 Td[(1)+nXi=k+1minfk,dig)]TJ /F8 7.97 Tf 26.81 14.95 Td[(kXi=1di>k(k)]TJ /F5 11.955 Tf 11.95 0 Td[(1)+T)]TJ /F8 7.97 Tf 6.59 0 Td[(kXi=kk+k)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xi=Bi+B)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xi=12)]TJ /F8 7.97 Tf 18.18 14.95 Td[(kXi=1(T)]TJ /F7 11.955 Tf 11.96 0 Td[(k+1)=k(T)]TJ /F7 11.955 Tf 11.95 0 Td[(k)+(k)]TJ /F7 11.955 Tf 11.96 0 Td[(B)(k)]TJ /F5 11.955 Tf 11.95 0 Td[(1+B)=2+B(B)]TJ /F5 11.955 Tf 11.96 0 Td[(1))]TJ /F7 11.955 Tf 11.96 0 Td[(k(2T)]TJ /F7 11.955 Tf 11.96 0 Td[(k+1)=2=(B2)]TJ /F7 11.955 Tf 11.96 0 Td[(B)=2)]TJ /F7 11.955 Tf 11.96 0 Td[(kwhereT=e=andB=(e=2)1=+1.Notethat=>ln2(2=+1)since>3ln2and0<<1.Hence\004(e=2)1=+1\004(e=2)1=>e=2k,thatis,fD(k)>0.Case2:k>e==2fD(k+1)fD(k)+2k)]TJ /F5 11.955 Tf 11.96 0 Td[(2dk+1fD(k)...fD(e==2)>0 Corollary5. AnoptimalsubstructureproblemisalsoNP-hardonpower-lawgraphsforall0<<1ifitisNP-hardonsimplegeneralgraphs. 41

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Proof. AccordingtoTheorem 2.16 ,wecanembedanundirectedgraphG=(V,E)intoapower-lawgraphG0oflyingin(0,1)andofverticespolynomialtimeinthesizeofG.SincetheoptimizationproblemhasoptimalsubstructurepropertyandGisasetofmaximalconnectedcomponentsofG0,itsoptimumsolutionforthegraphGcanbecomputedeasilyfromanoptimalsolutionforG0.ThiscompletestheproofofNP-hardness. 2.7ApproximationAlgorithmsAsthecomputationalhardnessandinapproximabilityresultsofclassicoptimizationproblemshavebeenshownintheprevioussections,thedesignofapproximationalgorithmisstillofgreatinterestbutremainsopen.Inthissection,wefocusonaddressingthefollowingquestions:Canthepropertyofpower-lawdegreedistributionhelpustodesignaneffectivealgorithmframeworkforNP-hardoptimizationproblems?Howcanweprovideatheoreticalframeworkforanalyzingapproximationratiosoftheseproblemsusingthispower-lawdegreeproperty?Willtheseapproximationratioschangedramaticallyfordifferentexponentialfactors,i.e.inpower-lawgraphswithdifferentdensities?Weproposeanalgorithmframework,calledLow-DegreePercolation(LDP)framework,tosolvetheoptimizationproblemsinpower-lawnetworks,includingMIS,MDS,andMVCproblems.TheideaofLDPframeworktopercolatethegraphstartingfromagreatnumberoflow-degreenodesinapower-lawgraph,allowsustodevelopatheoreticalframework,whichcanbeusedtoanalysistheapproximationratiosviaprobabilitytheory.Inparticular,weapplythistheoreticalframeworktoshowtheapproximationratiosfortheseproblemsontwowell-knownrandompower-lawmodelsin[ 2 21 ].Atlast,numericalanalysisofourproposedapproachesnotonlyvalidatesourtheoreticalanalysisbutalsoillustratestheeffectivenessofourapproachesinpractice. 42

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2.7.1Low-DegreePercolation(LDP)AlgorithmFrameworkInthissection,weproposedanalgorithmframeworktosolveoptimizationproblemsbytakingadvantageofthedegreesequencepropertyinpower-lawgraphs.Asonecansee,themostfundamentalpropertyofpower-lawgraphsarethattheycontainagreatnumberoflow-degreenodes,whileonlyasmallnumberofhigh-degreenodes.Therefore,theideaofourproposedLow-DegreePercolation(LDP)algorithmframeworkistosortthenodesbytheirdegreeandpercolatethegraphfromthenodesoflowestdegree.Theprocesscontinuesinresidualgraphiterativelyuntilnomorenodes,whicharesurelyinoptimalsolution,canbedetected.Atlast,weapplyexistingapproximationapproachestodetectthesolutionintheremaininggraph.ForMDSandMVCproblems,asshowninAlgorithm 4 ,sincethenodeincidenttoanodeofdegree1certainlybelongstoanoptimalsolution,wepercolatethegraphbyaddingalltheneighborsofnodeswithdegree1ineachiteration.Untilnomorenodesofdegree1existsinresidualgraph,weapplyexistingapproximationalgorithmin[ 83 ]forMDS(or[ 52 ]forMVC)toobtainthesolutioninthisresidualgraph. Algorithm4:LDPAlgorithmforMDS/MVCProblems Input:Power-lawgraphGOutput:MDS(orMVC)S 1while9Nodesofdegree1do 2foreachNodevofdegree1do 3AdditsneighborN(v)intoS; 4RemovevfromG; 5end 6RemoveallnodesincidenttoSfromgraphG; 7end 8DeterminetheleftoverMDS(orMVC)inGusingexistingapproximationalgorithmin[ 83 ](or[ 52 ])andaddthemintoS; 9returnS; Ontheotherhand,Algorithm 5 showsthealgorithmforMIS.Inthiscase,thenodesofdegree1willbelongtotheoptimalsolution,andinthemeanwhile,itiscertainthattheirneighborscannotbeinoptimalsolutionanymore.Therefore,inordertoobtain 43

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MIS,weselectallnodesofdegree1intothesolutionineachiteration.Atlast,weapplytheapproximationalgorithmin[ 40 ]toobtaintheMISintheremaininggraph. Algorithm5:LDPAlgorithmforMISProblem Input:Power-lawgraphGOutput:MISS 1while9Nodesofdegree1do 2foreachNodevofdegree1do 3AddvintoS; 4RemovevandallitsneighborsN(v)fromG; 5end 6end 7DeterminetheleftoverMISinGusingexistingapproximationalgorithmin[ 40 ]andaddthemintoS; 8returnS; Here,wenotethatinaspecialcasethattwonodesofdegree1areconnected,theoptimalsolutionofMDS(orMVC,MIS)containseitheroneofthem. 2.7.2ApproximationRatioAnalysisInthissection,weshowtheapproximationratioanalysisofLDPAlgorithmsinbothstructuralandexpectedrandompower-lawnetworks.Todothis,werstprovideatheoreticalframework,usingLDPalgorithm,toanalyzetheapproximationratiobasedontheprobabilitythatanodedoesnotconnecttoanynodeofdegree1.Then,thisframeworkisappliedtoshowtheratioofoptimizationproblemsintwodifferentmodels. 2.7.2.1TheoreticalframeworkInthistheoreticalframework,astheconnectedcomponentofsize2istrivial,wemainlyfocusontheratioanalysisintherestpartofpower-lawgraphs.Tobeginwith,werstprovideaformalproofofthefollowingLemma 8 (SimilarargumentforCorollary 6 ),whichhasbeenbrieydiscussedtheLDPalgorithms. Lemma8. IntheoptimalsolutiontoMDSandMVC,ifwedonotconsiderthecaseofconnectedcomponentswithsize2,theredonotexistanynodesofdegree1andallnodesincidenttoatleastonenodeofdegree1areselected. 44

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Proof. Intheproof,letubeanodeofdegree1incidenttoanothervofarbitrarydegreelargerthan1,weconsiderseveralcases:(1)Ifneitheruandvisselectedinoptimalsolution,noneighborisselectforuanduisnotselectedaswell,thisleadstoaninfeasiblesolution;(2)Ifbothuandvareselected,itiseasytoseethatthesolutionisnomoreoptimal;(3)Ifuisselectedinsteadofv,wehavetoselectasetofnodestosatisfyvifvhasdegreenolessthan2;(4)Ifvisselectedinsteadofu,bothuandvarealreadysatised,whichmeansthesizeofthesolutionlessthanthesizeinasolutioncontainingu.Accordingtotheseobservations,theproofiscomplete. Corollary6. IntheoptimalsolutiontoMIS,ifwedonotconsiderthecaseofconnectedcomponentswithsize2,allnodesofdegree1andallnodesincidenttoatleastonenodeofdegree1areselected.Next,wedene(,,i)tobetheprobabilitythatanodevofdegreeinotincidenttoanynodesofdegree1inapower-lawgraphG(,).Ourpurposeistoanalyzetheapproximationratiobasedon(,,i)inthisgraphG(,).LetXuibearandomvariablethatanodeuofdegreeidoesnotconnecttoanynodesofdegree1.Then,wehaveXui=n1,u2D10,u62D1whereD1isasetofnodesincidenttoatleastonenodeofdegree1.Notethatforallnodesofthesamedegree,theyhavethesamerandomvariables.Forsimplicity,wedeneXitobearandomvariablethatsomenodeofdegreei.Therefore,wehavetheexpectedvalueofnodeunotincidenttoanynodesofdegree1asE(Xi)=(,,i)Sincethenumberofnodesofdegreeiisequaltoe=i,byletting=e=andX=Pi=2e iXi,wehavethefollowinglemma: 45

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Lemma9. Theexpectednumberofnodesofdegreenolessthan2notincidenttoanynodesofdegree1isXi=2e i(,,i) Proof. Theexpectednumberofnodesnotincidenttoanynodesofdegree1isthesumofallnodesofdegreenolessthan2,i.e.X=Pi=2e iXi.ThenwehaveE(X)=Xi=2e iE(Xi)=Xi=2e i(,,i) Lemma10. ThevarianceofXisupperboundedbye2Xi=2Xj=2p (,,i)(,,j) (ij)where(,,i)=(,,i)(1)]TJ /F4 11.955 Tf 11.95 0 Td[((,,i)). Proof. Forarandomvariablecorrespondstoanodeofdegreeinotincidenttoanynodesofdegree1,thevarianceisVar(Xi)=1)]TJ /F7 11.955 Tf 33.18 8.09 Td[(i ()]TJ /F5 11.955 Tf 11.95 0 Td[(1)1)]TJ /F6 11.955 Tf 11.95 13.27 Td[(1)]TJ /F7 11.955 Tf 33.17 8.09 Td[(i ()]TJ /F5 11.955 Tf 11.96 0 Td[(1)=(,,i)(1)]TJ /F4 11.955 Tf 11.96 0 Td[((,,i))Foranytwovariablescorrespondtotwonodesofdegreeiandjnotincidenttoanynodesofdegree1,accordingtoCauchy-SchwarzInequality,wehavejCov(Xi,Xj)jp Var(Xi)Var(Xj)=p (,,i)(,,j)Then,wesumthemupandobtainXXi,XjjCov(Xi,Xj)jXXi,Xjp Var(Xi)Var(Xj)Xi=2e iXj=2e jp (,,i)(,,j)=e2Xi=2Xj=2p (,,i)(,,j) (ij) 46

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Therefore,wehavethevarianceofXtobeVar(X)=PXi,XjjCov(Xi,Xj)je2Pi=2Pj=2p (,,i)(,,j) (ij) Lemma11. Thenumberofnodesofdegreenolessthan2whichisnotincidenttoanynodesofdegree1islargerthanPi=2e iwithprobabilityatmost1 Pi=21 i)]TJ /F12 7.97 Tf 6.59 0 Td[((,,i)2 Pi=2Pj=2p (,,i)(,,j) (ij)+1 Proof. Let=Pi=2e i,accordingtoOne-SidedChebyshevInequality,Pr[X]=PrhX)]TJ /F7 11.955 Tf 11.95 0 Td[(E(X))]TJ /F7 11.955 Tf 11.95 0 Td[(E(X) p Var(X)p Var(X)i1 ()]TJ /F8 7.97 Tf 6.59 0 Td[(E(X))2 Var(X)+11 Pi=21 i)]TJ /F12 7.97 Tf 6.59 0 Td[((,,i)2 Pi=2Pj=2p (,,i)(,,j) (ij)+1 Forsimplicity,wedenethefollowingpandobtaintheCorollary 7 .p=1 Pi=21 i)]TJ /F12 7.97 Tf 6.58 0 Td[((,,i)2 Pi=2Pj=2p (,,i)(,,j) (ij)+1 Corollary7. Thenumberofnodesofdegreenolessthan2incidenttoatleastonenodeofdegree1isatleast(1)]TJ /F4 11.955 Tf 11.96 0 Td[()Pi=2e iwithprobabilityatleast1)]TJ /F7 11.955 Tf 11.96 0 Td[(p.Then,basedonLemma 8 ,wederivethefollowingapproximationratiosofMDSandMVCinapower-lawgraphG(,): 47

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Theorem2.17(MainTheorem(MDS&MVC)). Inapower-lawgraphG(,),byusingAlgorithm 4 ,MDSandMVCcanbeapproximatedinto1+()]TJ /F5 11.955 Tf 11.96 0 Td[(1)withprobabilityatleast1)]TJ /F7 11.955 Tf 12.36 0 Td[(p,whereistheapproximationratioofMDS(orMVC)inAlgorithm[ 83 ](or[ 52 ])w.r.t.agraphofsizeatmostePi=21 i. Proof. Let`bethenumberofnodesincidenttodegree1insomepower-lawgraphG(,).Wehavetheapproximationratioas`+OPT `+OPT`+(Pi=21 i)]TJ /F4 11.955 Tf 11.95 0 Td[(`) `+Pi=21 i)]TJ /F4 11.955 Tf 11.95 0 Td[(`AccordingtoCorollary 7 ,wehave`Pi=21 i)]TJ /F4 11.955 Tf 11.98 0 Td[(Pi=21 iwithprobabilityatleast1)]TJ /F7 11.955 Tf 11.98 0 Td[(p.Theproofiscomplete. IntermsofMIS,wehavetheapproximationratioasfollows: Theorem2.18(MainTheorem(MIS)). Inapower-lawgraphG(,),byusingAlgorithm 5 ,MIScanbeapproximatedintoN+ePi=21 i N+1 ePi=21 iwithprobabilityatleast1)]TJ /F7 11.955 Tf 12.26 0 Td[(p,whereNisthenumberofnodeswithdegree1,istheapproximationratioofMISinAlgorithm[ 40 ]w.r.t.agraphofsizeatmostePi=21 i.TheproofisomittedduetoitssimilarityoftheproofinTheorem 2.17 .Next,wefocusonapplyingthisframeworkontoPLRGmodelandanalyzingtheapproximationratios. 2.7.2.2Power-lawrandomgraphInPLRGgraph,thestraightforwardcomputationofPLRG(,,i)isintractableduetothedifcultytocalculateallpossiblecombinations.Tothisend,weconsidereachcasethatthereareparticularnumberofconnectedcomponentsofsize2inPLRG.At 48

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last,theapproximationfactorscanbederivedfromthelawoftotalprobability.Intherestofthissubsection,weshowtheprobabilitytohaveconnectedcomponentofsize2inaPLRGgraphandeachPLRG(,,i)respectively,andapplythemtoobtaintheapproximationratios. Lemma12. TheprobabilityPr[C2=]thatthereareconnectedcomponentsofsize2inaPLRGgraphis)]TJ /F8 7.97 Tf 6.64 -4.38 Td[(w2(2)!!)]TJ /F8 7.97 Tf 6.74 -4.38 Td[(N)]TJ /F8 7.97 Tf 6.59 0 Td[(ww)]TJ /F9 7.97 Tf 6.58 0 Td[(2(w)]TJ /F5 11.955 Tf 11.95 0 Td[(2)! N!!=(N)]TJ /F5 11.955 Tf 11.96 0 Td[(2w)]TJ /F5 11.955 Tf 11.96 0 Td[(1+2)!!whereN=e()]TJ /F5 11.955 Tf 11.95 0 Td[(1),w=eisthesizeofnodesofdegree1. Proof. Inordertohaveconnectedcomponentofsize2,2mini-nodesareselectedrstfromallwnodesofdegree1.Moreover,thereare(2)]TJ /F5 11.955 Tf 12.34 0 Td[(1)!!possibilitiestomatchthese2mini-nodes.Sincethenumberofperfectmatchingf(n)fornmini-nodesis(n)]TJ /F5 11.955 Tf 11.95 0 Td[(1)!!,theprobabilitycanbecalculatedbysimplifyingthefollowingequation.Pr[C2=]=)]TJ /F8 7.97 Tf 6.65 -4.38 Td[(w2(2)!!)]TJ /F8 7.97 Tf 6.75 -4.38 Td[(N)]TJ /F8 7.97 Tf 6.59 0 Td[(ww)]TJ /F9 7.97 Tf 6.58 0 Td[(2(w)]TJ /F5 11.955 Tf 11.96 0 Td[(2)!f(N)]TJ /F5 11.955 Tf 11.96 0 Td[(2w+2) f(N) Lemma13. InaPLRGgraphG,ifthereareconnectedcomponentofsize2,theprobabilitythatanodevofdegreeinotincidenttoanynodesofdegree1isPLRG(,,i)=nQw)]TJ /F9 7.97 Tf 6.58 0 Td[(1k=0N)]TJ /F8 7.97 Tf 6.59 0 Td[(i)]TJ /F8 7.97 Tf 6.58 0 Td[(w)]TJ /F8 7.97 Tf 6.58 0 Td[(k N)]TJ /F8 7.97 Tf 6.58 0 Td[(w)]TJ /F8 7.97 Tf 6.58 0 Td[(k,IfN)]TJ /F7 11.955 Tf 11.96 0 Td[(i)]TJ /F7 11.955 Tf 11.96 0 Td[(w>w;0,otherwise.whereN=e()]TJ /F5 11.955 Tf 11.96 0 Td[(1))]TJ /F5 11.955 Tf 11.96 0 Td[(2,w=e)]TJ /F5 11.955 Tf 11.96 0 Td[(2isthesizeofnodesofdegree1. Proof. LetD1beasetofnodesincidenttoatleastonenodeofdegree1.Considerthatthewholemini-nodesarecomposedofthreesubsets,i.e.,inodescorrespondenttov,wnodescorrespondenttoallnodesofdegree1andallleftovernodes,whichisreferredtoasNiandNwandNnfNi[Nwgrespectively.WhenN)]TJ /F7 11.955 Tf 11.97 0 Td[(i)]TJ /F7 11.955 Tf 11.98 0 Td[(w
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fromNnfNi[Nwg.Pr[v62D1]=)]TJ /F8 7.97 Tf 5.48 -4.38 Td[(N)]TJ /F8 7.97 Tf 6.59 0 Td[(i)]TJ /F8 7.97 Tf 6.58 0 Td[(www!f(N)]TJ /F5 11.955 Tf 11.96 0 Td[(2w) )]TJ /F8 7.97 Tf 5.48 -4.38 Td[(N)]TJ /F8 7.97 Tf 6.59 0 Td[(www!f(N)]TJ /F5 11.955 Tf 11.95 0 Td[(2w)=w)]TJ /F9 7.97 Tf 6.59 0 Td[(1Yk=0N)]TJ /F7 11.955 Tf 11.95 0 Td[(i)]TJ /F7 11.955 Tf 11.95 0 Td[(w)]TJ /F7 11.955 Tf 11.95 0 Td[(k N)]TJ /F7 11.955 Tf 11.95 0 Td[(w)]TJ /F7 11.955 Tf 11.95 0 Td[(kwheref(n)=(n)]TJ /F5 11.955 Tf 11.96 0 Td[(1)!!,representingthenumberofperfectmatchingfornnodes. Theorem2.19. InaPLRGgraphG,byusingAlgorithm 4 ,MDSandMVCcanbeapproximatedinto1+()]TJ /F5 11.955 Tf 11.96 0 Td[(1)withprobabilityatleastbe=2cY=0Pr[C2=](1)]TJ /F7 11.955 Tf 11.95 0 Td[(p)wherep=1 ()]TJ /F10 5.978 Tf 5.75 0 Td[((,,2))2 (,,2)+1inwhich=+ Pi=21 i. Proof. Consideronecasethatthereareconnectedcomponentsinthepower-lawgraph.Thus,accordingtoTheorem 2.17 ,theprobabilitythattheapproximationratioissmallerthan1+()]TJ /F5 11.955 Tf 10.26 0 Td[(1)is1)]TJ /F7 11.955 Tf 10.26 0 Td[(pforp=1 ()]TJ /F10 5.978 Tf 5.76 0 Td[((,,2))2 (,,2)+1where=+ Pi=21 i.Therefore,accordingtothelawoftotalprobability,thetheoremfollowsbytakingintoaccountall,whichrangesfrom0uptobe=2c. ForMISproblem,theapproximationratiocanbeobtainedasN+ePi=21 i N+1 ePi=21 iwithprobabilityatleastQbe=2c=0Pr[C2=](1)]TJ /F7 11.955 Tf 12.36 0 Td[(p),wherep0=1 (0)]TJ /F10 5.978 Tf 5.76 0 Td[((,,2))2 (,,2)+1inwhich0=+ Pi=21 i.NumericalAnalysis Fig. 2-3 illustratestheperformanceofourLDPalgorithmsinrandompower-lawgraphs,alongwiththerelationbetweendifferentandthecorrespondingapproximationratios,fromboththeoreticalandpracticalperspectives. 50

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AERPLModel BSRPLModelFigure2-3. NumericalresultsofourLDPalgorithmsondifferent(=5):(1)Theoreticalresultsshowstheapproximationratioswithprobabilityatleast1)]TJ /F7 11.955 Tf 11.95 0 Td[(o(1).Asonecansee,ourLDPalgorithmscanobtaintheoptimalsolutionforalltheseproblemsaftergetslargerthan1.6and1.7inERPLandSRPLrespectively,whichcoverstherangeofinmostreal-worldnetworks[ 18 ].Fortheothersmallerexponentialfactors,wecanseethattheapproximationratiosarealittlebithigher,especiallyupto5forMDSandMISproblemsforSRPLmodel.However,theprobabilitiesthatthesetwoproblemscanobtaintheapproximationratioslessthan1.5usingLDPalgorithmsareatleast0.95(onlyalittlebitlowerthan1)]TJ /F7 11.955 Tf 11.96 0 Td[(o(1)).(2)ExperimentalresultsfurtherrevealsthatourLDPalgorithmscanachieveevenbettersolutionsthantheoreticalbounds.(Wetestson100casesandchoosetheaverage.)AsillustratedinFig. 2-3 ,theapproximationratiosofallMDS,MVC,MISproblemsisnolargerthan1.2and2.5evenwhen=1.3inERPLandSRPLmodelsrespectively. 2.8RelatedWorksManyexperimentalresultsonrandompower-lawgraphsgiveusabeliefthattheproblemsmightbemucheasiertosolveonpower-lawgraphs.Eubanketal.[ 32 ]showedthatasimplegreedyalgorithmleadstoa1+o(1)approximationfactoronMINIMUMDOMINATINGSET(MDS)andMINIMUMVERTEXCOVER(MVC)onpower-lawgraphs(withoutanyformalproof)althoughMDSandMVChasbeenprovedNP-hardtobeapproximatedwithin(1)]TJ /F4 11.955 Tf 13.18 0 Td[()lognand1.366ongeneralgraphsrespectively[ 28 ].In[ 73 ],Gopalalsoclaimedthatthereexistsapolynomialtimealgorithmthatguaranteesa1+o(1)approximationoftheMVCproblemwithprobabilityatleast1)]TJ /F7 11.955 Tf 12.19 0 Td[(o(1).Unfortunately,thereisnosuchformalproofforthisclaimeither.Furthermore, 51

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severalpapersalsohavesometheoreticalguaranteesforsomeproblemsonpower-lawgraphs.Gkantsidisetal.[ 36 ]provedtheowthrougheachlinkisatmostO(nlog2n)onpower-lawrandomgraphswheretheroutingofO(dudv)unitsofowbetweeneachpairofverticesuandvwithdegreesduanddv.In[ 36 ],theauthorstakeadvantageofthepropertyofpower-lawdistributionbyusingthestructuralrandommodel[ 2 2 ]andshowthetheoreticalupperboundwithhighprobability1)]TJ /F7 11.955 Tf 12.31 0 Td[(o(1)andthecorrespondingexperimentalresults.Likewise,Jansonetal.[ 48 ]gaveanalgorithmthatapproximatedMAXIMUMCLIQUEwithin1)]TJ /F7 11.955 Tf 12.81 0 Td[(o(1)onpower-lawgraphswithhighprobabilityontherandompoissonmodelG(n,)(i.e.thenumberofverticeswithdegreeatleastidecreasesroughlyasn)]TJ /F8 7.97 Tf 6.58 0 Td[(i).Althoughtheseresultswerebasedonexperimentsandvariousrandommodels,theyraiseaninterestininvestigatinghardnessandinapproximabilityofoptimizationproblemsonpower-lawgraphs.Recently,Ferranteetal.[ 35 ]hadaninitialattemptonpower-lawgraphstoshowtheNP-hardnessofMAXIMUMCLIQUE(CLIQUE)andMINIMUMGRAPHCOLORING(COLORING)(>1)byconstructingabipartitegraphtoembedageneralgraphintoapower-lawgraphandNP-hardnessofMVC,MDSandMAXIMUMINDEPENDENTSET(MIS)(>0)basedontheiroptimalsubstructureproperties. 52

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CHAPTER3VULNERABILITYASSESSMENTInthischapter,usingthewell-knownrandompower-lawgraphmodel(SRPL)in[ 2 ],wedidanin-depthanalysisusingprobabilitytheorywithrespecttodifferentkindsofthreats:randomfailures,preferentialattacksanddegree-centralityattacks.Oursignicantconclusionsare(1)Acomplexnetworkcantoleraterandomfailuresifitsexponentialfactorislessthan2.9,(2)Power-lawnetworksaremorerobustunderpreferentialnodeattacksanddegree-centralitynodeattackswhentheyhavesmallerexponentialfactor,and(3)Inordertomaintainareliablecomplexsystem,weoptimizethepower-lawnetworksbyinvestigatingontheoptimalrangeofexponentialfactorbeforehand.Forbothcommunicationnetworksandsocialnetworks,thebestisillustratedtobelyingintheinterval[1.8,2.5],whichgivesadecentexplanationtothestructuresofreal-worldnetworks[ 4 12 34 74 ].When<1.8,themaintenanceofnetworkisverycostly,andwhen>2.5,thenetworkvulnerabilityisunpredictableduetoitsdependenceonthespecicattackingstrategy.(3)Whencascadingfailuresoccur,power-lawnetworksbecomeextremelyvulnerablewhenthefailurescanbepropagatedmorethan2hops. 3.1MetricOneofthemostcrucialquestioniswhichmeasurecopeswiththenetworkvulnerabilitythebest?Therehavebeenmanystudiesproposingdifferentmetricstoaccountforthenetworkvulnerability[ 3 5 60 63 ],amongwhichthedegreeofsuspectednodesoredges[ 5 ],theaverageshortestpathlength[ 3 ],theglobalclusteringcoefcients[ 60 ],theavailablenumberofcompromiseds)]TJ /F7 11.955 Tf 12.35 0 Td[(tows[ 63 ],thediameters,therelativesizeofthelargestclusterandtheaveragesizeoftheisolatedclusters[ 5 ]appeartobethemostpopularandeffective.Unfortunately,thesementionedmeasuresdonotseemtocastwellforsomeparticularkindsofnetworkvulnerabilities,especiallywhennetworkfragmentationisofhighpriority,asdepictedinFigure1. 53

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LetusconsiderasimpleexampleinFigure 3-1 illustratingasmallportionoftheInternet,wherenodesv1,v2,...,v7areISPsandtherestareconsumersortransmissionnodes.Asrevealedinthisgure,anysuccessfulcorruptiveattackstonodesv8andv10aresufcienttobringthewholenetworkdowntoitskneeswithnosatisedcustomers.Inadifferentattackingstrategy,theremovalofnodev7orv9,iftheadversarywastousemaximumdegreecentralityasthemetric,doesnotappeartoharmthenetworkfunctionbecauseallcustomersarestillsatised.Theseremovalsalsoreducetheglobalclusteringcoefcientsto0andincreasetheaverageshortestpathtonearly3.Besides,iftheattackerusestheavailablenumberofcompromisedowsfromv1tov2,thedestructionsofnodesv4andv7willdroptheowto1,andtheystill,unfortunately,cannotdestroytheexistenceofthegiantISPcomponentprovidingservicestothe(almost)wholenetwork. Figure3-1. AnExampleofInternet:theremovalofv8andv10(greynodes)issufcienttodestroythefunctionofthewholenetworksuchthatonlylessthan40%nodesconnecteachother. Thisexampleillustratesanimportantpointthattheothermetricsarelackof:Inordertobreakdownthenetwork,weneedtosomehowcontrolthebalanceamongdisconnectedcomponentswhileensuringthenonexistenceofgiantcomponents.Onepossibleandeffectivewaytodosoistomeasurethetotalpairwiseconnectivity(P),i.e.thenumberofconnectednode-pairs[ 27 ]inthenetwork.Backtoourexample,ascrutinylookintothedestructionsofnodesv8andv10,whichweknowcanbreak 54

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downthenetworkfunction,revealsthatthey,indeed,reducethenetworktotalpairwiseconnectivitytoitsgreatestextent(morethan60%).Thisgreatreduction,asaresult,signicantlymalfunctionsthewholenetwork.ThemeasurePalsolendsitselfeffectivelyalotofpracticalnetworkapplications.Aswediscussedabove,sincemanylarge-scalenetworkshavebeenshowntobepower-lawnetworks,theremovalofcriticalnodesandlinksregardingthismetricnotonlyreducesthenetworkperformancebutalsocanpossiblydisconnectthosenetworksfromtheoutsideworld.Anotherapplicationofthismetriccanbefoundindestroyingterroristnetworks,e.g.tobreakdownthecommunicationbetweenanytwoterroristindividualstothegreatestextent,aswellasprotectingthefunctionalityincommunicationnetworks. 3.2ThreatTaxonomyandNotationsIntherestofthischapter,wefocusoninvestigatingthevulnerabilityofpower-lawnetworksunderrandomfailuresorintentionalattacks.Thissectionconsistsofthefollowingparts:(1)threattaxonomy,includingrandomfailuresandintentionalattacks,and(2)usefulnotations. 3.2.1ThreatTaxonomyInthispaper,wefocusoninvestigatingtherobustnessofpower-lawnetworksunderrandomfailureandtwotypesofintentionalattacks,i.e.preferentialattackanddegree-centralityattack. Denition19(RandomFailure). EachnodeinG(,)failsrandomlywiththesameprobability. Denition20(PreferentialAttack). EachnodeinG(,)isattackedwithhigherprobabili-tyifithasahigherdegree. Denition21(Degree-CentralityAttack). Theadversaryonlyattacksthesetofdegree-centralitynodesinG(,). Denition22(RandomCascadingFailures). EachnodeinG(,)failsrandomlywiththesameprobability,andthefailurescanbecascaded. 55

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3.2.2NotationExplanationWithrespecttoeachthreat,wedenetheresidualnetworksofthepower-lawnetworkG(,)asGr,GpandGcaftertheoccurrenceofrandomfailure,preferentialattackanddegree-centralityattackrespectively.Theircorrespondingexpecteddegreesequencesaredenotedas~dr,~dpand~dc,wherethenumberofdri,dpianddciarereferredtoasyri,ypiandyci.Inaddition,wedeneapower-lawnetworkundercertainthreatstobehighly-connectedifa.s.itspairwiseconnectivityP=(n2)andlowly-connectedotherwise. 3.3PreliminariesInthissection,werstpresentsomeusefulresultsintheliterature,whichillustratetheimportantrelationsbetweenthesizeoflargestconnectedcomponentsandthedegreesequenceinrandomnetworks.Basedonthem,wethenderivesomefundamentalresultstoevaluatetherobustnessofpower-lawnetworks.Inthispaper,thesizeofaconnectedcomponentSGisthetotalnumberofnodesinSandtheconnectedcomponentSiscalledgiantcomponentifitssizeis(n). 3.3.1PreviousWorks Lemma14(M.MolloyandB.Reed[ 68 ]). InarandomgraphGwithinnodesofdegreeiwherePi=1i=1forthemaximumdegree, Q=nXi=1i(i)]TJ /F5 11.955 Tf 11.95 0 Td[(2)i(3)isametricwhichcanbeappliedtodeterminewhetherthereisgiantcomponentsinG.ThegiantcomponentsexistifQ>0and
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ofdegreesinaconnectedcomponent)atmostp nlognwithprobabilityatleast1)]TJ /F7 11.955 Tf 12.08 0 Td[(o(1)if~d<1.Heretheexpectedaveragedegree dandsecond-orderaveragedegree~daredenedas d=1 nnXi=1di,~d=Pni=1d2i Pni=1di(3)wherediistheelementsinthedegreesequence. Corollary8. Allconnectedcomponentsa.s.havesizesatmost1 2p nlogn+1if~d<1. Proof. ConsideraconnectedcomponentS,thevolumeofSisdenedasVol(S)=Pvi2Sdi.SincethereareatleastjSj)]TJ /F5 11.955 Tf 18.91 0 Td[(1edgesinaconnectedcomponentofsizejSj,wehave2(jSj)]TJ /F5 11.955 Tf 18.72 0 Td[(1)Vol(S)p nlogn.Therefore,thesizeofSisupperboundedby1 2p nlogn+1. 3.3.2RobustnessofIntactPower-lawNetworks Theorem3.1. Forapower-lawnetworkrepresentedasa(,)graphG(,), If<3.47875,thepairwiseconnectivityPis(n2); If3.47875,therangeofpairwiseconnectivityPisa.s.atmost1 2nc()n2 logn)]TJ /F5 11.955 Tf 11.96 0 Td[(1.wherec()=16=h()2)]TJ /F12 7.97 Tf 13.15 5.48 Td[(()]TJ /F9 7.97 Tf 6.59 0 Td[(2) ()]TJ /F9 7.97 Tf 6.59 0 Td[(1)i2isaconstantonanygiven.ToproveTheorem 3.1 ,werstshowtherelationbetweenthelargestcomponentandourmetric,thetotalpairwiseconnectivity,inthefollowinglemma. Lemma16. SupposethatthemaximumsizeofaconnectedcomponentinthegraphG=(V,E)is`,thepairwiseconnectivityPisthenatmostn(`)]TJ /F9 7.97 Tf 6.58 0 Td[(1) 2. Proof. Toprovetheupperbound,weconsidertheworstcasethatthewholenetworkconsistsofallconnectedcomponentsofsize`exceptsomeleftovernodes.Supposethattherearec1connectcomponentsofsize`andthenumberofleftovernodesisc2,wehaven=c1`+c2.Therefore,thepairwiseconnectivityPisPc1`2+c22c1`2+c2 ``2=c1`+c2 ``2=n(`)]TJ /F5 11.955 Tf 11.95 0 Td[(1) 2 57

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ProofofTheorem 3.1 : Proof. First,accordingtoF.Chungetal.[ 2 ],wecanndthethreshold3.47875ofsuchthatQ>0when<3.47875andQ<0when>3.47875.When<3.47875,accordingtoLemma 14 ,sinceQ>0,thereexistsonegiantcomponentofsize(n).Therefore,thepairwiseconnectivityPis(n2).When>3.47875,accordingtoAielloetal.[ 2 ],aconnectedcomponentSinthe(,)grapha.s.hasthesizeatmostc()n2 logn.ThentheupperboundofPfollowsdirectlyfromLemma 16 Inthefollowingthreesections,sincethepower-lawnetworkswithatleast3.47875arelowly-connectedeveniftheyarenotattacked,wewillfocusonexploitingtherobustnessofpower-lawnetworkswithlessthan3.47875underrandomfailures,preferentialattacksanddegree-centralityattacksrespectively. 3.4RandomFailuresInthissection,wefocusontherobustnessofpower-lawnetworksafterrandomfailures,inwhicheachnodehasthesameprobabilityp(0
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ToproveTheorem 3.6 ,werstshowtheexpecteddegreedistributioninGrasfollows. Lemma17. TheexpecteddegreedistributionofgraphGrisE(yri)=(1)]TJ /F7 11.955 Tf 11.95 0 Td[(p)i+1Xk=ikie kpk)]TJ /F8 7.97 Tf 6.59 0 Td[(iwheredegreeiis1i. Proof. LetpikbetheprobabilitythatanodevofdegreekinG(,)hasitsdegreetobeiinGr.Whenk1foranypand.Inanalternativeway,weuseLemma 14 andbranchingprocessmethodtoproveourtheorem.Thebasicideaisasfollows:accordingtotheexpecteddegreeofGr,werstndathresholdpusingLemma 14 ,whichdetermineswhetherthetotalpairwiseconnectivityPoftheresidualnetworkGrisa.s.(n2)ornot.Ifnot,thatis,>p,wefurtherusebranchingprocessmethodtoprovethatPinGrisa.s.atmost 59

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1 2ncr()n2 logn)]TJ /F5 11.955 Tf 11.96 0 Td[(1.First,wecomputepforGras Qr=Xi=1i(i)]TJ /F5 11.955 Tf 11.95 0 Td[(2)(1)]TJ /F7 11.955 Tf 11.96 0 Td[(p)i+1Xk=ikie kpk)]TJ /F8 7.97 Tf 6.59 0 Td[(i (3a)=e(1)]TJ /F7 11.955 Tf 11.96 0 Td[(p)Xi=11 iiXj=1j(j)]TJ /F5 11.955 Tf 11.96 0 Td[(2)ijpi)]TJ /F8 7.97 Tf 6.58 0 Td[(j(1)]TJ /F7 11.955 Tf 11.96 0 Td[(p)j (3b)=e(1)]TJ /F7 11.955 Tf 11.96 0 Td[(p)2Xi=1i2(1)]TJ /F7 11.955 Tf 11.96 0 Td[(p))]TJ /F7 11.955 Tf 11.96 0 Td[(i(2)]TJ /F7 11.955 Tf 11.96 0 Td[(p) i (3c).=e(1)]TJ /F7 11.955 Tf 11.96 0 Td[(p)2[(1)]TJ /F7 11.955 Tf 11.95 0 Td[(p)()]TJ /F5 11.955 Tf 11.96 0 Td[(2))]TJ /F5 11.955 Tf 11.96 0 Td[((2)]TJ /F7 11.955 Tf 11.96 0 Td[(p)()]TJ /F5 11.955 Tf 11.96 0 Td[(1)] (3d)wherestep( 3c )followssimilarlyfromthecalculationofexpectedvalueandvarianceinbinomialdistribution.Letusconsiderthecasethatthethresholdpsatises(1)]TJ /F7 11.955 Tf 10.99 0 Td[(p)()]TJ /F5 11.955 Tf 10.98 0 Td[(2))]TJ /F5 11.955 Tf 10.99 0 Td[((2)]TJ /F7 11.955 Tf 10.99 0 Td[(p)()]TJ /F5 11.955 Tf -458.7 -23.9 Td[(1)=0.When0.Thus,theexpectedpairwiseconnectivityE(P)isa.s.(n2)accordingtoLemma 14 Algorithm6:BranchingProcessMethod 1i 0; 2E0=L0=fvgbypickinganarbitrarynodev; 3whilejLij6=0do 4i i+1; 5ChooseanarbitraryufromLi)]TJ /F9 7.97 Tf 6.59 0 Td[(1andexposeallitsneighborsN(u); 6Ei=Ei)]TJ /F9 7.97 Tf 6.58 0 Td[(1[N(u); 7Li=(Lin(fug)[(N(u)nEi)]TJ /F9 7.97 Tf 6.58 0 Td[(1); 8end When>p,weusethefollowingbranchingprocessmethod(Algorithm 6 )onGraccordingtoitsexpecteddegreesequenceE(yri).Inthealgorithm,wedeneEiandLiasthesetofexposednodesandlivenodesiniterationirespectively,wherelivenodesarereferredtoasthesubsetofexposednodeswhoseneighborshavenotbeenexposed.NotethatjLij=0ifandonlyiftheentirecomponentisexposed.Forsimplicity,wedenerandomvariablesEi=jEijandLi=jLijasthenumberofexposednodesandlivenodes.LetTdenotethewholenumberofiterationsinbranchingprocess,thatis,T 60

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alsomeasuresthesizeofconnectedcomponentsinceexactlyonenodeisexposedineachiteration.WefurtherdeneanedgetobeabackedgeifitconnectsuandsomenodeinEi)]TJ /F9 7.97 Tf 6.58 0 Td[(1.WedenoteDi=jN(u)jandBi=jN(u)\Ei)]TJ /F9 7.97 Tf 6.58 0 Td[(1j)]TJ /F5 11.955 Tf 18.46 0 Td[(1measuringthedegreeofthenodeexposediniterationiandthenumberofbackedge.Bydenition,wehaveLi)-222(Li)]TJ /F9 7.97 Tf 6.58 0 Td[(1=Di)]TJ /F7 11.955 Tf 11.96 0 Td[(Bi)]TJ /F5 11.955 Tf 11.96 0 Td[(2immediately.Then,wecalculateE(Di),E(Bi)andE(Li)respectively.ConsideroneedgeinoriginalgraphG(,).Itstillexistsiffbothendpointsarenotfailed,thatis,theexpectednumberofedgesinGris(1)]TJ /F7 11.955 Tf 11.95 0 Td[(p)2m.Therefore,E(Di)=Xi=1ii(1)]TJ /F7 11.955 Tf 11.95 0 Td[(p)i+1Pk=i(ki)e kpk)]TJ /F17 5.978 Tf 5.76 0 Td[(i (1)]TJ /F7 11.955 Tf 11.95 0 Td[(p)2m=1 ()]TJ /F5 11.955 Tf 11.95 0 Td[(1)Xi=1i2(1)]TJ /F7 11.955 Tf 11.96 0 Td[(p)+ip i.=(1)]TJ /F7 11.955 Tf 11.96 0 Td[(p)()]TJ /F5 11.955 Tf 11.95 0 Td[(2) ()]TJ /F5 11.955 Tf 11.95 0 Td[(1)+pSincejN(u)\Ei)]TJ /F9 7.97 Tf 6.59 0 Td[(1j1,wehaveE(Bi)0.BysubstitutingE(Di)andE(Bi)intoLi)-222(Li)]TJ /F9 7.97 Tf 6.58 0 Td[(1=Di)]TJ /F7 11.955 Tf 11.96 0 Td[(Bi)]TJ /F5 11.955 Tf 11.96 0 Td[(2,wehaveE(Li)=L1+iXj=2E(Lj)-222(Lj)]TJ /F9 7.97 Tf 6.58 0 Td[(1)=d0+iXj=2E(Dj)]TJ /F7 11.955 Tf 11.96 0 Td[(Bj)]TJ /F5 11.955 Tf 11.95 0 Td[(2)d0+(i)]TJ /F5 11.955 Tf 11.96 0 Td[(1)(1)]TJ /F7 11.955 Tf 11.96 0 Td[(p)()]TJ /F5 11.955 Tf 11.96 0 Td[(2) ()]TJ /F5 11.955 Tf 11.96 0 Td[(1)+p)]TJ /F5 11.955 Tf 11.95 0 Td[(2=d0)]TJ /F4 11.955 Tf 11.95 0 Td[((p,)(i)]TJ /F5 11.955 Tf 11.95 0 Td[(1)where(p,)=2)]TJ /F7 11.955 Tf 11.95 0 Td[(p)]TJ /F5 11.955 Tf 11.95 0 Td[((1)]TJ /F7 11.955 Tf 11.95 0 Td[(p)()]TJ /F9 7.97 Tf 6.59 0 Td[(2) ()]TJ /F9 7.97 Tf 6.59 0 Td[(1)andtheinitialnodeisassumedtohavedegreed0.SincejLj)-222(Lj)]TJ /F9 7.97 Tf 6.58 0 Td[(1j=e ,accordingtoAzumaMartingaleInequality[ 22 ],Pr[jLi)]TJ /F7 11.955 Tf 11.96 0 Td[(E(Li)j>T]2eT2 2ie2 61

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wherei=16 ((p,))2e2 logn=cr()n2 lognandT=(p,)i=2.SinceweknowE(Li)+Td0)]TJ /F4 11.955 Tf 11.95 0 Td[((p,)(i)]TJ /F5 11.955 Tf 11.96 0 Td[(1)+(p,) 2i<0foranyd0.Therefore,PrT>16 ((p,))2e2 logn=Pr[T>i]Pr[Li>0]Pr[Li>E(Li)+T)]2eT2 2ie2 =2 n2Thus,theprobabilitythat,ingraphGr,thereisanon-failurenodevbelongingtoaconnectedcomponentofsizelargerthancr()n2 lognisatmostn2 n2=o(1),i.e.Grhasthelargestconnectedcomponentofsizea.s.cr()n2 logn.Hence,theupperboundofpairwiseconnectivityinGrfollowsfromLemma 16 directly. 3.4.2GoodRangeofunderRandomFailuresAccordingtoTheorem 3.6 ,weexploitthegoodrangeofexponentialfactorintermsofthepairwiseconnectivityPofpower-lawnetworks.Byobtainingthethresholdpfrom(1)]TJ /F7 11.955 Tf 12.21 0 Td[(p)(p)]TJ /F5 11.955 Tf 12.21 0 Td[(2))]TJ /F5 11.955 Tf 12.21 0 Td[((2)]TJ /F7 11.955 Tf 12.21 0 Td[(p)(p)]TJ /F5 11.955 Tf 12.21 0 Td[(1)=0,therelationbetweenthresholdpandfailureprobabilitypcanberevealedinthefollowingFig. 3-2 Figure3-2. RelationbetweenThresholdpandFailureProbabilityp 62

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BasedonTheorem 3.6 ,power-lawnetworksarehighly-connectedwhen>pandlowly-connectedotherwise.AsonecanseefromFig. 3-2 ,power-lawnetworksofexponentialfactor>2.9willstillremainhighly-connectedunderrandomfailuresevenwhenthefailureprobabilitypisunrealistically0.8.Thatis,wecancondentlyclaimthatpower-lawnetworkshaveanextremelyhightolerancetorandomfailureswhenitsexponentialfactor<2.9. 3.5PreferentialAttacksAspower-lawnetworksaretolerabletorandomfailures,onewillquestionwhetheritcanstilltolerateintentionalattacksiftheintrudersintendmoretoattackhubnodes.Inthissection,wefocusontherobustnessofpower-lawnetworksunderpreferentialattacks.Aswedenedabove,inpreferentialattacks,eachnodeinthenetworkisattackedwithhigherprobabilityifithaslargerdegree.Therefore,considerthecoststoattackforintruders,wefocusonthefollowingtwopreferentialattackschemes: InteractivePreferentialAttacks:onewaytocontrolthecoststoattackistoattackanodew.r.t.itsdegreeandanewparameter0.Thatis,anodeofdegreeiisattackedwithprobability1)]TJ /F9 7.97 Tf 16.1 4.71 Td[(1 i0; ExpectedPreferentialAttacks:anotherwaytocontrolthecoststoattackisbasedontheexpectednumberofnodesctoattack.Whentheintruderdecidesc,rangingbetween0ande(),anodeofdegreeiisattackedwithprobabilitypi=ci e()]TJ /F9 7.97 Tf 6.59 0 Td[(1)sincetheexpectednumberoffailurenodesisequaltoc,namelyPie ipi=c.Asonecansee,inboththeseschemes,anodeofhigherdegree,oftenreferredtoasahub,ismorepreferentiallyattacked,thatis,ithashigherprobabilitytobeattacked.BydenotingtheircorrespondingresidualgraphsasGIpandGEp,theirtotalpairwiseconnectivityareproveninTheorem 3.3 andTheorem 3.4 respectively. 3.5.1InteractivePreferentialAttackspi=1)]TJ /F9 7.97 Tf 16.1 4.7 Td[(1 i0 Theorem3.3. InaresidualgraphGIpofG(,)afterinteractivepreferentialattacks, If+0<3.47875,theexpectedpairwiseconnectivityE(P)is(n2); 63

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If+03.47875,thepairwiseconnectivityPisa.s.atmost1 2nc()n2 logn)]TJ /F5 11.955 Tf 11.95 0 Td[(1.wherec()=16=h()2)]TJ /F12 7.97 Tf 13.15 5.48 Td[(()]TJ /F9 7.97 Tf 6.59 0 Td[(2) ()]TJ /F9 7.97 Tf 6.59 0 Td[(1)i2isaconstantonanygiven.Theorem 3.3 canbeproveninthesamewayasinTheorem 3.1 aftershowingtheexpecteddegreeinresidualgraphGIpasinLemma 20 ,whichisbasedonthefollowingtwolemmas. Lemma18. IngraphG(,),theprobabilitythatanodevofdegreeiincidenttoanothernodeuofdegreexisix e()]TJ /F9 7.97 Tf 6.59 0 Td[(1). Proof. Consideranodevofdegreei,inthematchingofmini-nodes,atleastoneofimini-nodesforvconnectstoanotheroneofxfornodeuofdegreex.Thus,wehave)]TJ /F8 7.97 Tf 6.36 -4.38 Td[(i1)]TJ /F8 7.97 Tf 10.96 -4.38 Td[(x1f(N)]TJ /F5 11.955 Tf 11.95 0 Td[(2) f(N)=ix N)]TJ /F5 11.955 Tf 11.96 0 Td[(1=ix N+O(1 N2).=ix e()]TJ /F5 11.955 Tf 11.95 0 Td[(1)wheref(n)=(n)]TJ /F5 11.955 Tf 12.33 0 Td[(1)!!representingthenumberofperfectmatchingsforNnodesandN=e()]TJ /F5 11.955 Tf 11.96 0 Td[(1)denotesthenumberofmini-nodes. Lemma19. Foranodevofdegreei,theexpectednumberofnon-failureneighborsE(NIp(i))ofvisi(+0)]TJ /F9 7.97 Tf 6.59 0 Td[(1) ()]TJ /F9 7.97 Tf 6.58 0 Td[(1). Proof. AccordingtoLemma 18 ,nodevhasprobabilityix e()]TJ /F9 7.97 Tf 6.59 0 Td[(1)toconnecttonodeuofdegreex.Sincenodeuofdegreexhasthenon-failureprobability1 x0,thenwehavetheexpectednon-failureneighborofvtobeE(NIp(i)).=Xx=1ix e()]TJ /F5 11.955 Tf 11.95 0 Td[(1)1 x0e x.=i(+0)]TJ /F9 7.97 Tf 6.58 0 Td[(1) ()]TJ /F9 7.97 Tf 6.59 0 Td[(1)Theprooscomplete. Lemma20. TheexpecteddegreedistributionofgraphGIpisE(ypi).=e i()]TJ /F9 7.97 Tf 6.59 0 Td[(1) (+0)]TJ /F9 7.97 Tf 6.58 0 Td[(1)+0wherei2n(+0)]TJ /F9 7.97 Tf 6.58 0 Td[(1) ()]TJ /F9 7.97 Tf 6.59 0 Td[(1),...,(+0)]TJ /F9 7.97 Tf 6.58 0 Td[(1) ()]TJ /F9 7.97 Tf 6.59 0 Td[(1)o. 64

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Proof. ConsiderthesetofnodeswithdegreeiinGp,theyarecorrespondenttothenodesofdegreexintheoriginalgraphG(,).Hence,theexpectedunattackednodesinthissetise x1 x0=e x+0.AccordingtoLemma 19 ,weknowtherelationbetweeniandxisi.=x(+0)]TJ /F9 7.97 Tf 6.59 0 Td[(1) ()]TJ /F9 7.97 Tf 6.58 0 Td[(1).Therefore,wehavetheexpectednumberofnodesofdegreeiinGIptobee i()]TJ /F16 5.978 Tf 5.75 0 Td[(1) (+0)]TJ /F16 5.978 Tf 5.76 0 Td[(1)+0. 3.5.2ExpectedPreferentialAttackspi=ci e()]TJ /F9 7.97 Tf 6.59 0 Td[(1) Theorem3.4. InaresidualgraphGEpofG(,)afterexpectedpreferentialattacks, ThepairwiseconnectivityPisa.s.(n2)ifc1; ThepairwiseconnectivityPisa.s.atmost1 4n3 2lognifc>maxc1)]TJ /F8 7.97 Tf 17.63 5.48 Td[(c()]TJ /F9 7.97 Tf 6.59 0 Td[(2) e()]TJ /F9 7.97 Tf 6.59 0 Td[(1)2()]TJ /F9 7.97 Tf 6.59 0 Td[(2))]TJ /F17 5.978 Tf 10.37 3.86 Td[(c()]TJ /F16 5.978 Tf 5.75 0 Td[(3) e()]TJ /F16 5.978 Tf 5.75 0 Td[(1) ()]TJ /F9 7.97 Tf 6.59 0 Td[(1))]TJ /F17 5.978 Tf 10.37 3.86 Td[(c()]TJ /F16 5.978 Tf 5.75 0 Td[(2) e()]TJ /F16 5.978 Tf 5.75 0 Td[(1)<1.ToproveTheorem 3.4 ,weagainrstshowtheexpecteddegreedistributioninGEpasfollows. Lemma21. Foranodevofdegreei,theexpectednumberofnon-failureneighborsE(NEp(i))ofvisi1)]TJ /F8 7.97 Tf 17.64 5.48 Td[(c()]TJ /F9 7.97 Tf 6.58 0 Td[(2) e()]TJ /F9 7.97 Tf 6.59 0 Td[(1)2. Proof. AccordingtoLemma 18 ,thenodevhasprobabilityix e()]TJ /F9 7.97 Tf 6.59 0 Td[(1)toconnectnodeuofdegreex.Sincenodeuofdegreexhasthenon-failureprobability1)]TJ /F7 11.955 Tf 11.99 0 Td[(cx e()]TJ /F9 7.97 Tf 6.58 0 Td[(1),thenwehavetheexpectednon-failureneighborofvtobeE(Np(i)).=Xx=1ix e()]TJ /F5 11.955 Tf 11.95 0 Td[(1)1)]TJ /F7 11.955 Tf 35.16 8.09 Td[(cx e()]TJ /F5 11.955 Tf 11.96 0 Td[(1)e x.=i1)]TJ /F7 11.955 Tf 18.45 8.09 Td[(c()]TJ /F5 11.955 Tf 11.96 0 Td[(2) e()]TJ /F5 11.955 Tf 11.96 0 Td[(1)2Theproofiscomplete. Corollary9. TheexpecteddegreedistributionofgraphGEpisE(ypi).=e i1)]TJ /F7 11.955 Tf 18.45 8.09 Td[(c()]TJ /F5 11.955 Tf 11.96 0 Td[(2) e()]TJ /F5 11.955 Tf 11.96 0 Td[(1)20@1)]TJ /F7 11.955 Tf 81.66 8.09 Td[(ci (e()]TJ /F5 11.955 Tf 11.95 0 Td[(1))1)]TJ /F8 7.97 Tf 17.63 5.47 Td[(c()]TJ /F9 7.97 Tf 6.59 0 Td[(2) e()]TJ /F9 7.97 Tf 6.59 0 Td[(1)21A 65

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wherei2n1)]TJ /F8 7.97 Tf 17.63 5.48 Td[(c()]TJ /F9 7.97 Tf 6.59 0 Td[(2) e()]TJ /F9 7.97 Tf 6.59 0 Td[(1)2,...,1)]TJ /F8 7.97 Tf 17.63 5.48 Td[(c()]TJ /F9 7.97 Tf 6.59 0 Td[(2) e()]TJ /F9 7.97 Tf 6.58 0 Td[(1)2o.ProofofTheorem 3.4 : Proof. Intheproof,werstcalculatetheexpectedaveragedegree yEpbasedonCorollary 9 as dEp.=Px=1e x1)]TJ /F8 7.97 Tf 27.42 4.7 Td[(cx e()]TJ /F9 7.97 Tf 6.59 0 Td[(1)x1)]TJ /F8 7.97 Tf 17.63 5.47 Td[(c()]TJ /F9 7.97 Tf 6.58 0 Td[(2) e()]TJ /F9 7.97 Tf 6.58 0 Td[(1)2 n)]TJ /F7 11.955 Tf 11.95 0 Td[(candsecond-orderaveragedegree~dEpas~dEp.=Px=1e x1)]TJ /F8 7.97 Tf 27.41 4.71 Td[(cx e()]TJ /F9 7.97 Tf 6.58 0 Td[(1)x1)]TJ /F8 7.97 Tf 17.63 5.48 Td[(c()]TJ /F9 7.97 Tf 6.59 0 Td[(2) e()]TJ /F9 7.97 Tf 6.59 0 Td[(1)22 Px=1e x1)]TJ /F8 7.97 Tf 27.42 4.71 Td[(cx e()]TJ /F9 7.97 Tf 6.59 0 Td[(1)x1)]TJ /F8 7.97 Tf 17.63 5.48 Td[(c()]TJ /F9 7.97 Tf 6.59 0 Td[(2) e()]TJ /F9 7.97 Tf 6.58 0 Td[(1)2.=1)]TJ /F7 11.955 Tf 18.44 8.08 Td[(c()]TJ /F5 11.955 Tf 11.95 0 Td[(2) e()]TJ /F5 11.955 Tf 11.96 0 Td[(1)2()]TJ /F5 11.955 Tf 11.96 0 Td[(2))]TJ /F8 7.97 Tf 15.72 5.48 Td[(c()]TJ /F9 7.97 Tf 6.59 0 Td[(3) e()]TJ /F9 7.97 Tf 6.59 0 Td[(1) ()]TJ /F5 11.955 Tf 11.96 0 Td[(1))]TJ /F8 7.97 Tf 15.72 5.47 Td[(c()]TJ /F9 7.97 Tf 6.59 0 Td[(2) e()]TJ /F9 7.97 Tf 6.59 0 Td[(1)AccordingtoLemma 15 andCorollary 8 ,thereexistsonegiantcomponentif yEp>1andallcomponentshavesizeatmost1 2p nlogn+1if~yEp<1,thentheprooffollowsfromLemma 16 directly. 3.5.3RelationsbetweenandExpectedAttackedNodesIninteractivepreferentialattacks,accordingtoTheorem 3.3 ,apower-lawnetworkswithexponentialfactorwillbelowly-connectediftheintruderselecta0suchthat+03.47875.Sinceanodeofdegreeiisattackedwithprobability1)]TJ /F9 7.97 Tf 16.71 4.7 Td[(1 i0inthisscheme,thisnodecansurvivewithprobability1 i0.Therefore,wehavetheexpectednumberofsurvivednodesasXie i1 i0.=e(+0)thatis,theexpectedpercentageofattackednodescanbeobtainedbycalculating1)]TJ /F12 7.97 Tf 13.15 5.47 Td[((+0) ().Fig. 3-3 reportstherelationbetweenandexpectedattackednodesunderiterativepreferentialattacks.Weobservedthattheexpectednumberofattackednodes 66

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Figure3-3. RelationbetweenandAttackedNodesunderIterativePreferentialAttacks decreasessharplywiththeincreaseof.Clearly,smallerleadstoamorerobustpower-lawnetwork. Figure3-4. RelationbetweenandAttackedNodesunderExpectedPreferentialAttacks Inexpectedpreferentialattacks,againFig. 3-4 revealsthesmallerthebetter.AccordingtoTheorem 3.4 ,exceptofuncertainareas(shadowareas),wecanseethatthepercentageofattackednodes(undertheredline)reduceswhenincreases. 3.6Degree-CentralityAttacksAspower-lawnetworksisquitevulnerableunderpreferentialattacks,theirtolerationtothedeterministicintentionalattacksattractsmoreattentions.Also,onecanalso 67

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questionwhetheritisstilltrueunderdeterministicintentionalattacksthatpower-lawnetworkswithsmallercanbettermaintaintheirfunctionalities.Inthissection,weconsiderthedegree-centralityattack,inwhichtheintrudersintentionallyattackthehubs,thesetofnodesofhighestdegrees.Whenallnodesofdegreelargerthanx0areattackedsimultaneously,wehavethefollowingTheorem 3.5 3.6.1RobustnessunderDegree-CentralityAttacks Theorem3.5. InaresidualgraphGcofG(,)afterdegree-centralityattacks, ThepairwiseconnectivityPisa.s.(n2)ifx0>min(x01 ()]TJ /F9 7.97 Tf 6.58 0 Td[(1)Px0x=11 x)]TJ /F16 5.978 Tf 5.76 0 Td[(12 Px0x=11 x>1); ThepairwiseconnectivityPisa.s.atmost1 4n3 2lognifx0
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ProofofTheorem 3.5 : Proof. Withtheexpectedaveragedegree ycas dc=Px0x=1e xx1 ()]TJ /F9 7.97 Tf 6.59 0 Td[(1)Px0i=11 i)]TJ /F16 5.978 Tf 5.76 0 Td[(1 n)]TJ /F6 11.955 Tf 11.96 8.97 Td[(Pi=x0+1e i=1 ()]TJ /F5 11.955 Tf 11.96 0 Td[(1))]TJ 5.47 -.72 Td[(Px0x=11 x)]TJ /F16 5.978 Tf 5.76 0 Td[(12 Px0x=11 xandsecond-orderaveragedegree~ycas~dc=Px0x=1e xhx1 ()]TJ /F9 7.97 Tf 6.59 0 Td[(1)Px0i=11 i)]TJ /F16 5.978 Tf 5.75 0 Td[(1i2 Px0x=1e xx1 ()]TJ /F9 7.97 Tf 6.59 0 Td[(1)Px0i=11 i)]TJ /F16 5.978 Tf 5.75 0 Td[(1=1 ()]TJ /F5 11.955 Tf 11.95 0 Td[(1)x0Xx=11 x)]TJ /F9 7.97 Tf 6.59 0 Td[(2TherestofproofisthesameasTheorem 3.4 3.6.2RelationsbetweenandAttackedNodesFig. 3-5 illustratestherelationsbetweenandattackednodesunderdegree-centralityattacksbasedonTheorem 3.5 .Ontheonehand,itissimilartoexpectedpreferentialattacksthatthepercentageofattackednodes(undertheredline)reduceswhenincreasesexceptofuncertainareas(shadowareas).Ontheotherhand,underdegree-centralityattacks,theintruderonlyneedstoattackroughly8%lessnumberofnodestolowerdownthepairwiseconnectivityofpower-lawnetworksthanunderexpectedpreferentialattacks. Figure3-5. RelationbetweenandAttackedNodesunderDegree-CentralityAttacks 69

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3.7RandomCascadingFailuresMoreimportantly,thefailureswillleadtoamuchmoredevastatingconsequenceespeciallywhenfailuresarecascaded,i.e.,thesefailednodescancausetheoverloadandfailureoftheirnearbyelementsinthesystembecauseoftheloadshifting.LetusconsideranexampleinFigure 3-6 illustratingasmallportionofthepowergrid,wherenodesv1,v2,...,v7aregenerators,v8,v9,...,v16aretransmittersandv17,v18,v19arecustomers.Eachnodehasloadequaltoitsdegreeandcapacityequaltotwiceitsdegree.Asrevealedinthisgure,anysuccessfulcorruptiveattackstonodesv8andv10canaffectthepowersupplyfromgeneratorsortransmittersinstantly,whilecustomersarestillabletoutilizetheelectricityuntiltheleftelectricityindemandcentersisusedup.However,whenfailuresarecascaded,alltransmitterscanfailsequentially(gradualcolorchanges),i.e.,v8,v10)v12)v9,v15)v11,v13)v14,v16,leadingtonopowersupplytocustomersinstantly.Therefore,inordertocontinuouslymaintainthenormalnetworkfunctions,itisofgreatimportancetoassessthenetworkvulnerabilityinthepresentofcascadingfailures,beforehand.Inthissection,takingintoaccountcascadingfailures,weanalyzethenetworkvulnerabilityviatwomainthrusts,complexnetworkstructureanalysisandoptimaldetectionofmostvulnerablenodes,basedontherecentlyproposedeffectivemetric,to-talpairwiseconnectivity[ 27 77 78 ].Bymeasuringtheconnectednode-pairsinresidualnetworks(apairofnodesareconnectedwhenthereisafunctionalpathbetweenthem),theminimumoftotalpairwiseconnectivitycancontrolthebalanceamongdisconnectedcomponentsandfurtherensurethenonexistenceofgiantcomponents,leadingtothedestructionofnetworkfunctionality. 3.7.1CascadingFailureModelInthispaper,weuseoneofthemostwell-acceptedmodelsproposedin[ 85 ],inwhicheachnodeuinthenetworkhasathresholdu2[0,1],typicallydrawnfromsomeprobabilitydistribution.StartingwithaninitialsetoffailurenodesF0,called 70

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Figure3-6. Eachnodeinthispowergridhasloadequaltoitsdegree,capacityequaltotwiceitsdegreeandeachredarrowsaystheshiftingof2unitload.Thesolidredarrowsstandforthedirectfailurecausedbythecascadesandthedottedonesmeantheloadshiftingtotheneighborwhichisnotfaileddirectly.Theoverloadandfailureofv8andv10canonlycausethedisconnectionfromgeneratorsandtransmitters,yetthepowercanbestillsuppliedtocustomersfromdemandcenters.However,whenfailurecascades,itleadstothebreakdownofalltransmittersandtheelectricitytocustomersareaffectedinstantly. vulnerablenodes,thedynamicsoffailurecascadesunfoldroundbyroundasfollows.Thecascadingprocessisdeterministicallyindiscreterounds:inroundt,allnodesthatfailedinroundt)]TJ /F5 11.955 Tf 12.76 0 Td[(1remainfailed,andanothernodevfailsifthetotalnumberofitsfailureneighborsisatleastu,i.e.,jN(u)\Ft)]TJ /F9 7.97 Tf 6.58 0 Td[(1judeg(u),inwhichFt)]TJ /F9 7.97 Tf 6.58 0 Td[(1isthesetoffailurenodesbeforeroundt)]TJ /F5 11.955 Tf 11.95 0 Td[(1.Inaddition,thismodelisalsoexhibitedasoneofthetwomaincascadingmodelsinthecontextofsocialscienceliterature,whichisreferredtoasLinearThresholdPropagationmodel.Moreimportantly,thismodelbelongstothecategoryofmostcontagionproblems,suchasmodelsoffailuresinengineeringsystems,i.e.,powergrid[ 75 ],theInternet[ 5 ],ormodelsofepidemics[ 53 ]andsoon.Intheliterature,someoftheworksassumethatthethresholdsuaregivenasapartoftheinput.However,thethresholdsareusuallygivenasconstantincommunicationnetworksduetothexedloadofeachnode.Ontheotherhand,theyare 71

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generallynotavailableandnon-trivialtoinferinsocialnetworks[ 39 ].Therefore,insteadoftherandomthreshold,weuseasimpliedvariationinwhichanodefailsifafractionofitsneighborsfailedinthepreviousround. 3.7.2CascadingRandomFailuresInordertoanalyzethepairwiseconnectivityintheresidualpower-lawgraph,werstprovidethefollowingtheorembyextendingfromTheorem 3.6 : Theorem3.6. ConsidertheresidualgraphG0withexpecteddegreesequence~d=fd1,d2,...,dng(~y=fy1,y2,...,y0grepresentsthenumberofelementswithvalueiin~d)andmaximumdegree0.Giventhefollowingconditionsw.r.t.theboundsofrst-orderdegreesummation d1min(G0)nXi=1di= dnd1max(G0)(3)andtheboundsofthesecond-orderdegreesummation d2min(G0)nXi=1d2i=0Xj=1j2yjd2max(G0)(3)Then,wehavethepairwiseconnectivityisa.s.atmost1 2n)]TJ /F7 11.955 Tf 5.48 -9.69 Td[(c20logn)]TJ /F5 11.955 Tf 11.96 0 Td[(1.wherec=16 2)]TJ /F17 5.978 Tf 7.79 4.46 Td[(d2max(G0) d1min(G0).Notethattheboundsd1min(G0)andd2max(G0)aremoreimportanttoassessthepairwiseconnectivitywhentheresidualnetworkisfragmented. Proof. HereweonlyshowthedifferentpartsfromtheproofofTheorem 3.6 .Afterbranchingprocessmethod,weagainfocusoncalculatingE(Di),E(Bi)andE(Li)respectively.Notethatwewilljustfocusonthedifferentstepsfrom[ 78 ]inthisproof.Let=2)]TJ /F8 7.97 Tf 13.15 5.7 Td[(d2max(G0) d1min(G0).Wehave,E(Di)d2max(G0) d1min(G0)=2)]TJ /F4 11.955 Tf 11.95 0 Td[(Similarasin[ 78 ],wehaveE(Bi)0duetojN(u)\Ei)]TJ /F9 7.97 Tf 6.59 0 Td[(1j1.ThenE(Li)d0)]TJ /F4 11.955 Tf 11.96 0 Td[((i)]TJ /F5 11.955 Tf 11.95 0 Td[(1) 72

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isderivefromthesubstitutionofE(Di)andE(Bi)intoLi)-222(Li)]TJ /F9 7.97 Tf 6.59 0 Td[(1=Di)]TJ /F7 11.955 Tf 11.96 0 Td[(Bi)]TJ /F5 11.955 Tf 11.96 0 Td[(2,wheretheinitialnodeisassumedtohavedegreed0.SincethemaximumdegreeintheresidualgraphG0is0,wehavejLj)-233(Lj)]TJ /F9 7.97 Tf 6.58 0 Td[(1j0.AccordingtoAzuma0sMartingaleInequality[ 22 ],Pr[jLi)]TJ /F7 11.955 Tf 11.95 0 Td[(E(Li)j>]2e)]TJ /F16 5.978 Tf 5.76 0 Td[(2 2ie20wherei=16 220logn=c20lognand= 2i.SinceE(Li)+d0)]TJ /F4 11.955 Tf 12.07 0 Td[((i)]TJ /F5 11.955 Tf 12.07 0 Td[(1)+ 2i<0foranyd0,wehavePrT>2e)]TJ /F16 5.978 Tf 5.75 0 Td[(2 2i202 n2Then,theprobabilitythatthereisanon-failurenodebelongingtoaconnectedcomponentofsizelargerthanc20logningraphG0isatmosto(1)andtheupperboundpairwiseconnectivityinG0followsfromLemma 16 directly. Inthissubsection,wefocusoninvestigatingtheexpecteddegreesequenceoftheresidualgraph,alongwithitsupperandlowerboundsofrstandsecondorderdegreesummation. Lemma23. Whenp>,theupperboundmaxfPrkdgoftheprobabilityPrkdthatanodevofdegreeksurvivesafterd>0roundcascadescanberecursivelycomputedusing(1)]TJ /F7 11.955 Tf 11.95 0 Td[(p))]TJ /F5 11.955 Tf 6.67 -1.6 Td[(1 2exp)]TJ /F5 11.955 Tf 11.95 0 Td[(2k)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F4 11.955 Tf 11.95 0 Td[()]TJ /F9 7.97 Tf 17.64 14.95 Td[(Xi=1imaxfPrid)]TJ /F9 7.97 Tf 6.59 0 Td[(1g2wherei=1 i)]TJ /F16 5.978 Tf 5.75 0 Td[(1()]TJ /F9 7.97 Tf 6.58 0 Td[(1)istheprobabilitythatoneofaneighborforanarbitrarynodehasanodeofdegreei[ 78 ],andPri0=1)]TJ /F7 11.955 Tf 12.27 0 Td[(pforanydegreeisinceeachnoderandomlyfailswiththesameprobabilitypatthebeginning. 73

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Proof. Consideranodevofdegreekandtheprobabilityithatvhasaneighborofdegreei.WehavePr[vhasxineighborsofdegreei]=k! Qi=1xi!Yi=1ixiForaneighborofvwithdegreei,itcouldeitherfailinroundjwithprobabilitypijorsurviveafterdroundswithprobabilityqi(d)]TJ /F9 7.97 Tf 6.58 0 Td[(1)(=Prid)]TJ /F9 7.97 Tf 6.59 0 Td[(1),thatis,Pd)]TJ /F9 7.97 Tf 6.59 0 Td[(1j=0pij+qi(d)]TJ /F9 7.97 Tf 6.59 0 Td[(1)=1.Letfijbetheneighborsofdegreeifailedinroundjandsi(d)]TJ /F9 7.97 Tf 6.59 0 Td[(1)betheneighborsofdegreeisurvivedafterdroundcascades.Notethattheprobabilitiespijandqi(d)]TJ /F9 7.97 Tf 6.59 0 Td[(1)canbederivedfromthepower-lawrandomnetworkmodelonlybasedonthedegreeiofanodeinaparticularroundj,alongwiththeinitialfailureprobabilitypofeachnode.Therefore,wehavePr[vsurvivesatround0\fijneighborsofdegreeifailinroundj\si(d)]TJ /F9 7.97 Tf 6.59 0 Td[(1)neighborsofdegreeisurviveafterroundd)]TJ /F5 11.955 Tf 11.96 0 Td[(1]=XxineighborsofdegreeiPr[vsurvivesatround0\fijneighborsoutofxiofdegreeifailinroundj\si(d)]TJ /F9 7.97 Tf 6.59 0 Td[(1)neighborsoutofxiofdegreeisurviveafterroundd)]TJ /F5 11.955 Tf 11.96 0 Td[(1jvhasxineighborsofdegreei]Pr[vhasxineighborsofdegreei]=(1)]TJ /F7 11.955 Tf 11.95 0 Td[(p)Yi=1)]TJ /F7 11.955 Tf 37.41 -1.6 Td[(xi! Qd)]TJ /F9 7.97 Tf 6.59 0 Td[(1j=1fij!si(d)]TJ /F9 7.97 Tf 6.59 0 Td[(1)d)]TJ /F9 7.97 Tf 6.58 0 Td[(1Yj=0pfijijqsi(d)]TJ /F16 5.978 Tf 5.76 0 Td[(1)i(d)]TJ /F9 7.97 Tf 6.59 0 Td[(1)k! Qixi!Yi=1ixi=(1)]TJ /F7 11.955 Tf 11.96 0 Td[(p)k! QiQjfij!Qisi(d)]TJ /F9 7.97 Tf 6.58 0 Td[(1)Yi=1d)]TJ /F9 7.97 Tf 6.59 0 Td[(1Yj=0(ipij)fijYi=1(iqi(d)]TJ /F9 7.97 Tf 6.59 0 Td[(1))si(d)]TJ /F16 5.978 Tf 5.76 0 Td[(1)wherethethirdstepholdssincetheprobabilityisequalto0whenthereexistssomexi6=Pd)]TJ /F9 7.97 Tf 6.59 0 Td[(1j=0fij+si(d)]TJ /F9 7.97 Tf 6.59 0 Td[(1).Also,itiscleartoseethatPi=1Pd)]TJ /F9 7.97 Tf 6.59 0 Td[(1j=0ipij+Pi=1iqi(d)]TJ /F9 7.97 Tf 6.59 0 Td[(1)=1. 74

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Accordingtothecascadingmodel,nodevsurvivesafterdhopcascadesifandonlyiflessthanfractionofitsneighborsfailafterd)]TJ /F5 11.955 Tf 11.95 0 Td[(1roundcascades.Therefore,wehavePrkd=XPi=1Pd)]TJ /F16 5.978 Tf 5.76 0 Td[(1j=0fijk(1)]TJ /F7 11.955 Tf 11.95 0 Td[(p)k! QiQjfij!Qisi(d)]TJ /F9 7.97 Tf 6.58 0 Td[(1)Yi=1d)]TJ /F9 7.97 Tf 6.58 0 Td[(1Yj=0(ipij)fijYi=1(iqi(d)]TJ /F9 7.97 Tf 6.58 0 Td[(1))si(d)]TJ /F16 5.978 Tf 5.75 0 Td[(1)=(1)]TJ /F7 11.955 Tf 11.96 0 Td[(p)XPi=1Pd)]TJ /F16 5.978 Tf 5.76 0 Td[(1j=0fijkkPi=1Pd)]TJ /F9 7.97 Tf 6.59 0 Td[(1j=0fij(Xi=1d)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xj=0ipij)PiPjfij(1)]TJ /F9 7.97 Tf 17.63 14.95 Td[(Xi=1d)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xj=0ipij)k)]TJ /F18 7.97 Tf 6.58 5.98 Td[(PiPjfij(1)]TJ /F7 11.955 Tf 11.96 0 Td[(p))]TJ /F5 11.955 Tf 6.68 -1.59 Td[(1 2exp)]TJ /F5 11.955 Tf 11.96 0 Td[(2k)]TJ /F9 7.97 Tf 13.15 5.26 Td[(Xi=1d)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xj=0ipij)]TJ /F4 11.955 Tf 11.96 0 Td[(2=(1)]TJ /F7 11.955 Tf 11.96 0 Td[(p))]TJ /F5 11.955 Tf 6.68 -1.6 Td[(1 2exp)]TJ /F5 11.955 Tf 11.96 0 Td[(2k)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F9 7.97 Tf 17.64 14.95 Td[(Xi=1iqi(d)]TJ /F9 7.97 Tf 6.59 0 Td[(1))]TJ /F4 11.955 Tf 11.95 0 Td[(21 2(1)]TJ /F7 11.955 Tf 11.95 0 Td[(p)exp)]TJ /F5 11.955 Tf 11.96 0 Td[(2k)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F4 11.955 Tf 11.96 0 Td[()]TJ /F9 7.97 Tf 17.63 14.94 Td[(Xi=1imaxfPrid)]TJ /F9 7.97 Tf 6.59 0 Td[(1g2wherethethirdstepfollowsfromtheHoeffding0sinequality[ 76 ]andthelaststepfollowsfrommaxfPrid)]TJ /F9 7.97 Tf 6.58 0 Td[(1g<1)]TJ /F7 11.955 Tf 12.01 0 Td[(psincetheprobabilityofthesurvivalofanodehastobesmallerthantheprobabilityitfailswithoutcascadingfailures. Lemma24. Whenp>,thelowerboundminfPrkdgoftheprobabilityPrkdthatanodevofdegreeksurvivesafterd>0roundcascadescanberecursivelycomputedusing(1)]TJ /F7 11.955 Tf 11.95 0 Td[(p)kk)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F9 7.97 Tf 17.63 14.95 Td[(Xi=1iminfPrid)]TJ /F9 7.97 Tf 6.58 0 Td[(1gk)]TJ /F9 7.97 Tf 13.15 5.26 Td[(Xi=1iminfPrid)]TJ /F9 7.97 Tf 6.58 0 Td[(1g(1)]TJ /F12 7.97 Tf 6.59 0 Td[()kwhereiistheprobabilitythatanodeofdegreekhasaneighborofdegreei,andPri0=1)]TJ /F7 11.955 Tf 12.17 0 Td[(pforanydegreeisinceeachnoderandomlyfailswiththesameprobabilitypatthebeginning. 75

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Proof. AccordingtotheproofofLemma 23 ,weknowthatPrkd=(1)]TJ /F7 11.955 Tf 11.95 0 Td[(p)XPi=1Pd)]TJ /F16 5.978 Tf 5.76 0 Td[(1j=0fijkkPi=1Pd)]TJ /F9 7.97 Tf 6.58 0 Td[(1j=0fij(Xi=1d)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xj=0ipij)PiPjfij(1)]TJ /F9 7.97 Tf 17.63 14.94 Td[(Xi=1d)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xj=0ipij)k)]TJ /F18 7.97 Tf 6.59 5.98 Td[(PiPjfij(1)]TJ /F7 11.955 Tf 11.95 0 Td[(p)kk(1)]TJ /F9 7.97 Tf 17.64 14.94 Td[(Xi=1iqi(d)]TJ /F9 7.97 Tf 6.59 0 Td[(1))k(Xi=1iqi(d)]TJ /F9 7.97 Tf 6.59 0 Td[(1))(1)]TJ /F12 7.97 Tf 6.59 0 Td[()kNext,considerthefunctionf(x)=(1)]TJ /F7 11.955 Tf 12.1 0 Td[(x)yxk)]TJ /F8 7.97 Tf 6.59 0 Td[(yforsome0x1and0yk.Wehavedf(x) dx=(1)]TJ /F7 11.955 Tf 11.95 0 Td[(x)y)]TJ /F9 7.97 Tf 6.59 0 Td[(1xk)]TJ /F8 7.97 Tf 6.59 0 Td[(y)]TJ /F9 7.97 Tf 6.59 0 Td[(1(k)]TJ /F7 11.955 Tf 11.95 0 Td[(kx)]TJ /F7 11.955 Tf 11.96 0 Td[(y)Itiseasytoseethatdf(x) dx>0iffk)]TJ /F7 11.955 Tf 12.17 0 Td[(kx)]TJ /F7 11.955 Tf 12.17 0 Td[(y>0.Thatis,x<1)]TJ /F4 11.955 Tf 12.17 0 Td[(wheny=k.SinceminPrid)]TJ /F9 7.97 Tf 6.59 0 Td[(1. Lemma25. Theexpectednumberofnodeofdegreek0=kPi=1iPridinresidualgraphcanbeestimatedase kPrkdwheretheboundsofPrkdisdeterminedbyLemma 23 and 24 Proof. ConsideranodeofdegreekinoriginalgraphG.Aftercascadingfailures,itsdegreecanbeestimatedbasedonthesurvivalofitsneighbors.Particularly,foreachneighbor,ithasprobabilityitoconnecttoanodeofdegreei.Moreover,anodeofiinGwillsurviveafterd-roundcascadingfailureswithprobabilityPrid.Therefore,foranodeofdegreekinG,itsdegreeintheresidualgraphcanbeestimatedaskPi=1iPrid.Ontheotherhand,eachnodeofdegreekinGhasprobabilityPrkdtosurviveaftercascadesandtherearee knodesofdegreekinG.Therefore,theproofiscomplete. Usingthelemma 25 ,wecanobtainthefollowingTheorem: Theorem3.7. Theexpectedrst-orderandsecond-orderdegreesummationaree ()]TJ /F5 11.955 Tf 11.96 0 Td[(1)Xi=1Xk=11 k)]TJ /F9 7.97 Tf 6.59 0 Td[(1i)]TJ /F9 7.97 Tf 6.59 0 Td[(1PridPrkd 76

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ande ()]TJ /F5 11.955 Tf 11.95 0 Td[(1)2Xi=1Xj=1Xk=11 k)]TJ /F9 7.97 Tf 6.59 0 Td[(2i)]TJ /F9 7.97 Tf 6.59 0 Td[(1j)]TJ /F9 7.97 Tf 6.58 0 Td[(1PridPrjdPrkdwheretheboundsaredeterminedbyminfPrkdgandmaxfPrkdg. Proof. Accordingtothedenitionofrst-orderdegreesummation,wehaveXi=1k0e kPrkd=Xi=1)]TJ /F7 11.955 Tf 5.48 -9.69 Td[(kXi=1iPride kPrkd=eXi=1Xi=11 i)]TJ /F9 7.97 Tf 6.59 0 Td[(1()]TJ /F5 11.955 Tf 11.96 0 Td[(1)1 k)]TJ /F9 7.97 Tf 6.58 0 Td[(1PridPrkd=e ()]TJ /F5 11.955 Tf 11.95 0 Td[(1)Xi=1Xk=11 k)]TJ /F9 7.97 Tf 6.59 0 Td[(1i)]TJ /F9 7.97 Tf 6.59 0 Td[(1PridPrkdAgain,accordingtothedenitionofsecond-orderdegreesummation,wehaveXi=1k0e kPrkd=Xi=1)]TJ /F7 11.955 Tf 5.48 -9.68 Td[(kXi=1iPrid2e kPrkd=e ()]TJ /F5 11.955 Tf 11.95 0 Td[(1)2Xi=1Xj=1Xk=11 k)]TJ /F9 7.97 Tf 6.59 0 Td[(2i)]TJ /F9 7.97 Tf 6.59 0 Td[(1j)]TJ /F9 7.97 Tf 6.59 0 Td[(1PridPrjdPrkd 3.7.3NumericalAnalysisHereweshowthatourtheoreticalanalysisconsistswellwiththesimulationresult.Particularly,wegeneratepower-lawnetworksusingigraphpackage[ 26 ]andtestonthesyntheticnetworkswithdifferentparameters,exponentialfactorandnetworksizen.Duetothesimilarresultsusingdistinctparameters,weonlyprovidetheresultwith=1.5andn=250asinFig. 3-7 .AsrevealedinFig. 3-7 ,apartfromthesurprisingagreementbetweenouranalysisandsimulation,wealsondthatpower-lawnetworks 77

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Figure3-7. NumericalAnalysisinPower-LawNetworks(=1.5,n=250).Weplotthethreecascadinghopsandndthatouranalysis(pinkplots)approximatesthesimulationofthetotalpairwiseconnectivity(PWC)aftercascadingfailuressurprisinglywell,inbothcasesthatpower-lawnetworksarea.s.unaffected(PWC/n2)anda.s.fragmented. arenolongerrobustunderrandomfailureswhencascadingfailuresoccur.Forexample,wheneachnodeisattackedwithprobabilityonly0.4,thenetworkisa.s.fragmentedonlyafter1-hoppropagation.Thistransitionhappensonlywhentheprobabilityequalto0.2iffailurescancascade2hopsandalmostvanisheswhenallowingmorehopcascades. 3.8RelatedWorksThereareagreatnumberofstudiesregardingthetoleranceofreal-worldnetworksagainstfailuresandattacksusingdifferentmetrics.EdgevulnerabilityinmetabolicnetworkswasstudiedbyKaiseretal.withrespecttotheaverageshortestpathandtheclusteringcoefcient[ 50 ].Forthesakeofpowergridnetworks,Albertetal.[ 3 ]investigatedtheirvulnerabilitybymeasuringthelossofconnectivityundervariousthreats,includingrandom,cascading,load-basedanddegree-basednodalfailures.ThedisruptionofvitalinterstatesystemswasassessedbyMatisziwetal.[ 63 ]accordingtotheavailablenumberofcompromiseds)]TJ /F7 11.955 Tf 12.95 0 Td[(tows.Cohenetal.[ 23 ]showedtheresilienceofInternettotherandombreakdownofthenodesbasedonpercolationtheory.In[ 72 ],Satorrasetal.alsorevealedthattherandomuniformimmunizationof 78

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individualscannotleadtotheeradicationofcommunicationsincomplexsocialnetworksusingthereducedprevalencerate.Doyleetal.[ 30 ]andSydneyetal.[ 81 ],usinganovelmetricELASTICITY,exploredthatInternettopologiesarelessaffectedbybothrandomandtargetedattacksthanthepower-lawnetworks.Ingeneral,therobustnessofothercomplexnetworkswasstudiedin[ 46 47 ]usingalgebraicconnectivity,i.e.,thesecond-smallesteigenvalueoftheLaplacianmatrixofagraph.Recently,fromadifferentperspective,Aldersonetal.[ 7 ]focusedontheroleoforganizationanddesignintermsofthecomplexityinhighlyorganizedtechnologicalandbiologicalsystems.Moregenerally,Albertetal.[ 5 ]rstcomparedtherobustnessofcomplexsystemswiththepower-lawandexponentialproperties.Bymeasuringthediameters,therelativesizeofthelargestclusterandtheaveragesizeoftheisolatedclusters,thepower-lawnetworksareempiricallyobservedtotoleratefailurestoasurprisingdegreebuttheirsurvivabilitydecreasesrapidlyunderattacksaftercomparingthemwithexponentialnetworks.Lateron,Holmeetal.[ 43 ]furtherinvestigatedthedegreeofharmstopower-lawnetworksunderdifferentstrategiesofattacks.Unfortunately,alltheseobservationsarederivedfromexperimentsandlacktheirtheoreticalfoundations. 79

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CHAPTER4OPTIMIZATIONOFPOWER-LAWNETWORKSInthischapter,weinvestigatethetradeoffimpactofmaintenancecostsandrobustnessguaranteeonthepower-lawnetworks.Inparticular,wefocusonthepower-lawnetworkswith<2.9,whichhavebeendiscoveredtotoleraterandomfailurestoanextremehighdegree.Inaddition,sinceFig. 3-3 3-4 and 3-5 alreadyrevealedthatpower-lawnetworkscantoleratepreferentialattacksiftheycantoleratedegree-centralityattackswhen<2.9,wefocusontheguaranteeoftheirfunctionalityunderdegree-centralityattacks.Westudythepracticalcommunicationnetworksandsocialnetworksrespectivelytoexploretheunderlyingreasonsoftheirreal-worldnetworktopologies.Ontheotherhand,weshowtheNP-hardnesstodetectthesecriticalelementsinpower-lawnetworks,alongwithtwoalgorithmsintwodifferentcases:elementfailuresandcascadingfailures.Theeffectivenessofouralgorithmsareevaluatedonbothsyntheticpower-lawnetworksandreal-worldnetworks. 4.1DesignOptimizationofPower-lawNetworksTheabovevulnerabilityassessmentsgiveusabeliefthatpower-lawnetworksaremorerobustwhenissmaller.However,amajorityofreal-worldnetworksusuallyhavetheirexponentialfactorrangingfrom2to2.5ratherthansomesmallapproaching1orevenless.Thequestionsareintuitivelyraised:Isitbetterifreal-worldnetworksaredensersuchthattheycanbemorerobust?Whatcausesthemtobesparserthanourexpectation?Doesthereexistsomepotentialoptimizationfactors?Toaddressthesequestions,inthissection,weinvestigatethetradeoffimpactofmaintenancecostsandrobustnessguaranteeonthepower-lawnetworks.Inparticular,wefocusonthepower-lawnetworkswith<2.9,whichhavebeendiscoveredtotoleraterandomfailurestoanextremehighdegree.Inaddition,sinceFig. 3-3 3-4 and 3-5 alreadyrevealedthatpower-lawnetworkscantoleratepreferentialattacksifthey 80

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cantoleratedegree-centralityattackswhen<2.9,wefocusontheguaranteeoftheirfunctionalityunderdegree-centralityattacks.Westudythepracticalcommunicationnetworksandsocialnetworksrespectivelytoexploretheunderlyingreasonsoftheirreal-worldnetworktopologies. 4.1.1CommunicationNetworksInthedesignofcommunicationnetworks,suchastheInternet,telecommunicationnetworksandsoon,wearerequirednotonlytoguaranteetheirfunctionalitybutreducethemaintenancecostsaswell.Amongvariousnetworkperformancemetric,i.e.,delay,packetloss,throughput,etc.,theguaranteeofitsconnectivityisofthehighpriority.Thatis,areal-worldnetworkonlyneedsufcientnumberoflinkstoguaranteeitsfunctionality,anditsotherperformancemetriccanbeguaranteedbyadjustingitscapacityplanning[ 59 ].Inparticular,weconsiderthecostsincludingthelinkcostsandtheprotectioncostsforcriticalnodes.Sincethenodeswithdegreeandbetweennesscentralityarecloselycorrelatedinnon-fractalpower-lawnetworks[ 56 ],hereweconsiderthecriticalnodestobedegree-centralitynodes.Toformulatetheoptimizationfunctionforpower-lawnetworksincommunicationnetworks,werstprovethefollowingLemma 26 byconsideringtheworstcasewithrespecttotherobustnessofpower-lawnetworks.Thatis,asmentionedabove,afterprotectingthedegree-centralitynodes,power-lawnetworksa.s.remainshighly-connected(itstotalpairwiseconnectivityisa.s.(n2))eventhoughallothernodesarefailed. Lemma26. LetGcpbetheresidualgraphofG(,)onlyconsistingoftheprotecteddegree-centralitynodes(thenodesofdegreelargerthanx0),wehave ThepairwiseconnectivityPisa.s.(n2)ifx01); ThepairwiseconnectivityPisa.s.atmost1 4n3 2lognifx0>minnx01 ()]TJ /F9 7.97 Tf 6.58 0 Td[(1)Px=x0+11 x)]TJ /F16 5.978 Tf 5.75 0 Td[(2<1o. 81

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Proof. Considerthatweprotectonlyallnodesofdegreelargerthanx0andallothernodesarefailed.SimilarasinCorollary 10 ,theexpecteddegreesequencecanbewrittenasE(ycpi).=e i 1 ()]TJ /F5 11.955 Tf 11.95 0 Td[(1)Xx=x0+11 x)]TJ /F9 7.97 Tf 6.59 0 Td[(1!TherestofproofisthesameasTheorem 3.4 Inordertoguaranteethefunctionalityofapower-lawnetwork,wetaketheaboveLemma 26 asthecondition.Inthemeanwhile,weaimtominimizethemaintenancecosts,whichincludethelinkcostsandcriticalnodeprotectioncosts.Indetail,weconsiderthefollowingcostfunctions: LinkCosts:Consideralink(u,v)inG(,),itslinkcostisheavilydependentonthenumberofmessagesittransmitsaccordingto[ 62 ].Anothercrucialfactorforthelinkcostisitscapacityow[ 79 ].Sincethenodeswithdegreeandbetweennesscentralityarecloselycorrelatedinnon-fractalpower-lawnetworks[ 56 ],weconsiderthelinkcostproportionaltotheaverageofthedegreesofitstwoendpoints. CriticalNodeProtectionCosts:IntermsofthecriticalnodesinG(,),apartfromtheirdegrees,theirprotectioncostsarealsocloselyrelatedwiththenetworkdensity.Accordingto[ 79 ],thecostswillrisewiththeincreaseofdensitysinceitenlargesthedemandofmessageexchanges.Inaddition,asinvestigatedin[ 62 ],thechainreactionleadstotheroughlyexponentialincreaseofcosts,weconsiderthecost(x)toprotectanodeofdegreexasaxb=forsomeconstantaandb.Therefore,wecancondentlyformulatethefollowingMixedLinearProgramming(MIP),withtwovariablesx0and,as min1 2Px=1e xx+Px=x0+1e x(x)s.t.1 ()]TJ /F9 7.97 Tf 6.58 0 Td[(1)Px=x0+11 x)]TJ /F16 5.978 Tf 5.75 0 Td[(12 Px=x0+11 x>1x02Z+,x0>0 (4) Notethatweomittheproportionalconstantoflinkcostsinceitdoesnotaffecttheoptimizationoftotalmaintenancecosts. 82

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4.1.2SocialNetworksAswementionedatthebeginningofthispaper,oneofthemainthreatsinsocialnetworksisthemalwarepropagations[ 87 ].Thus,apartfromthefactorsin[ 11 ],thecontainmentsofthesemaliciousspreadingbecomeanothercrucialfactorofthesparsicationofsocialnetworks.Inotherwords,whenanindividualisinfected,wewanttominimizetheexpectednumberoftotalinfectedusers,whichcanberealizedbyimmunizingcriticalusersbeforehand.Therefore,theminimizationofimmunizationcostsbecomesanurgentneed.Thus,inordertoformulatetheoptimizationfunctionforpower-lawnetworksinsocialnetworks,werstinvestigatetheupperboundofexpectedsizeofaconnectedcomponentafterprotectingthecriticalusers,whichareagainreferredtoasthedegree-centralitynodes.Thatis,wefocusonthesizeofconnectedcomponentsonresidualnetworkafterremovingsuchimmunizedusers.BydeningtheresidualgraphGstobetheresidualpower-lawgraphG[VnS]afterimmunizingindividualsinS,thefollowingTheorem 4.1 givestheboundofexpectedsizeofaconnectedcomponentonGs. Theorem4.1. IntheresidualgraphGsofG(,),theexpectedsizeofaconnect-edcomponentcisa.s.upperboundedbyOn1 4when~ds<1,thatis,x0
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Ontheotherhand,accordingtoF.Chungetal.[ 21 ],weknowthattheprobabilityforanyrandompairofnodesbelongingtothesamecomponentisupperboundedby~d2s (1)]TJ /F5 11.955 Tf 12.48 2.66 Td[(~ds)n dsCombiningtheabovetwobounds,weknow1 n2 d2sC2n~d2s (1)]TJ /F5 11.955 Tf 12.48 2.66 Td[(~ds)n dswhichimpliesthat ds~d2s C2(1)]TJ /F5 11.955 Tf 12.48 2.66 Td[(~ds)Thatis,bychoosingCtobelogn,withprobabilityatleast1)]TJ /F7 11.955 Tf 12.18 0 Td[(o(1),theexpectedsizecofconnectedcomponentsisa.s.atmostOn1 4. Again,considertheabovelemmaastheconditionandthesameprotectioncostfunctionofcriticalusers(x)=axb=,weformulatethefollowingmixedlinearprogramming,withtwovariablesx0and,inordertomakesurethattheexpectedsizeofconnectedcomponentsintheresidualpower-lawnetworksisnolargerthanOn1=4. minPx=x0+1e x(x)s.t.1 ()]TJ /F9 7.97 Tf 6.58 0 Td[(1)Px0x=11 x)]TJ /F16 5.978 Tf 5.76 0 Td[(2<1x02Z+,x0>0 (4) 4.1.3OptimalRangeofExponentialFactorForthesakeofcommunicationnetworks,considerthepracticalrangeofprotectioncostsfrom0tox9=foranodeofdegreex(thatisb2[0,3]),Fig. 4-1 revealstherelationbetweenmaintenancecostsandoptimalaccordingtoMIP( 4 ).Asonecansee,theoptimalisfrom1.8to2.5nomatterhowlargetheconstantbis,theexponentialfactorisnolessthan1.8.(Notethatthecurveisinvariantfordistinct 84

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networksizessincetheeffectoflight-tailedelementsinriemannzetafunctioncanbeneglected.) Figure4-1. OptimalRobustCommunicationNetworks Fig. 4-2 reportsthattheoptimalrangeofisfrom2.3to2.4insocialnetworksaccordingtoMIP( 4 ).Weobservethattheincreaseofbdoesnotreallyaffecttherangeofandthecurvealsoremainsinvariantwithrespecttodifferentnetworksizes. Figure4-2. OptimalRobustSocialNetworks Insummary,theanalysisonbothcommunicationnetworksandsocialnetworksgiveusareasonableexplanationofthetopologyinreal-worldpower-lawnetworks,thatis,thebestrangeoftheexponentialfactoris[1.8,2.5].Inothercases,thenetwork 85

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maintenancecosteitherbecomesveryexpensivewhen<1.8,orthenetworkrobustnessisunpredictablewhen>2.5duetoitsdependenceonthespecicattackingstrategy. 4.2CriticalElementsDetectioninPower-lawNetworksInthissection,westudytwopracticaloptimizationproblemsnamelyCriticalLinkandNodeDisruptor(CLDandCND),toassessthenetworkvulnerabilitywhenagivennumberofnetworkelements(linksornodes)failundesirably.Werefertotheseelementsascriticallinksandnodeshereafter. Denition23(CriticalLinkDisruptor). GivenanintegerkandaweightedundirectedgraphG=(V,E,W),theproblemasksforaweight-boundedsubsetofcriticallinksSE,i.e.P(i,j)2Swijk,whoseremovalminimizesthetotalpairwiseconnectivityinG[EnS]. Denition24(CriticalNodeDisruptor). GivenanintegerkandaweightedundirectedgraphG=(V,E,W),theproblemasksforaweight-boundedsubsetofcriticalnodesSV,i.e.Pvi2Swik,whoseremovalminimizesthetotalpairwiseconnectivityinG[VnS].Moreover,takingintoaccountthecascadingfailures,wedeneanotherproblemfocusingonthedetectionofcriticalnodes,calledCascadingVulnerabilityNodeDetec-tion(CVND)problem,asfollows: Denition25(CascadingCriticalNodeDisruptor). Giventwointegersk,d,afractionalnumber2(0,1)andanundirectedgraphG=(V,E).LetP(S)betotalpairwiseconnectivityofresidualgraphGafterthed-hopcascadingfailurescausedbytheinitialremovalofthesetofnodesS2V.TheCVNDproblemasksforkmostvulnerablenodessuchthatP(S)isminimized. 4.2.1HardnessofDetectingCriticalLinksandNodesInthissubsection,weshowthatCLDandCNDproblemsareNP-hard,whichdeniestheexistenceofapromptoptimalsolution. 86

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Lemma27(Ferranteetal.[ 35 ]). LetG1=(V1,E1)beasimplegraphwithnnodesand1.Formaxf4,logn+log(n+1)g,wecanconstructapower-lawgraphG=G1[G2withexponentialfactorandthenumberofnodese()byconstructingabipartiteG2asamaximalcomponentinG. Lemma28. Thecliqueseparator(CS)problem(whichisdenedasgivenanundi-rectedgraphG=(V,E),ndaminimumsetoflinksSEsuchthattheconnectedcomponentsofG[EnS]arecliques,eachhassizeatleast3)isNP-hard. Theorem4.2. TheCLDproblemisNP-hardonpower-lawgraphsevenifallnodeshaveunitweights. Proof. ConsiderthedecisionversionofCLDthataskswhetheranundirectedgraphG=(V,E)containsasetoflinksSEofsizeksuchthatthepairwiseconnectivityinresidualgraphG[EnS]isatmostcforagivenpositiveintegerc.ToprovethatCLDonpower-lawgraphsisinNP-hard,wereducethecliqueseparator(CS)toit.Afterconstructingapower-lawgraphG0=G[GbwherethebipartitegraphGb=(Ub,Vb;Eb)isamaximalcomponentinG0accordingtoLemma 27 ,weshowthatthereisaCSofsizekinGiffG0hasaCLDS0ofsizek0suchthatthepairwiseconnectivityofG0[E0nS0]isatmostc,wherek0=k+jEbj)-229(jMbjandc=jEj)]TJ /F7 11.955 Tf 18.09 0 Td[(k+jMbj.NotethatMbisthelinksinthemaximummatchingofGb.First,supposeSVisacliqueseparatorofGwithjSj=k.WehavethepairwiseconnectivityinG[EnS]tobejEj)]TJ /F7 11.955 Tf 14.36 0 Td[(ksinceallcomponentsinthisgrapharecliques.SincethemaximummatchingonGbcanbefoundinpolynomialtimeusingHopcroft-Karpalgorithm[ 44 ],thepairwiseconnectivityonG0iscafterremovingadditionaljEbj)-222(jMbj.Conversely,supposethatS0V0isaCLDofG0withsizek0.NotethatS0=A[Sb,whereAandSbareCLDonGandGbrespectively.WeshowthatthenumberofcriticallinksSbinGbisjEbj)-209(jMbj.IfjSbj
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removingtheCSk.IfjSbj>jEbj)-254(jMbj,thepairwiseconnectivityofGbreducebyonewhenremovingonemorelinkfromthemaximummatching.Meanwhile,alinkaddedontotheresidualgraphofGwillincreasethepairwiseconnectivityatleastoneifitconnectstwoindependentnodesandatleast3ifithasoneendpointbelongingtosomecomponentintheresidualgraphofG.Thus,wehaveSb=jEbj)-248(jMbjanditiseasytoverifythatAisaCSofG. Theorem4.3. TheCNDproblemisNP-hardonpower-lawgraphsevenifallnodeshaveunitweights. Proof. ConsiderthedecisionofCNDthataskswhetheragraphG=(V,E)containsasetofnodesSVofsizeksuchthatthepairwiseconnectivityinG[VnS]isatmostcforagivenpositiveintegerc.ToprovethatCNDonpower-lawgraphsisinNP-hard,wereducethevertexcover(VC)toit.LetanundirectedgraphG=(V,E)wherejVj=nandapositiveintegerknbeanyinstanceofVC.Weconstructapower-lawgraphG0=(V0,E0)asfollows.First,foreachnodevi2VongraphG,weaddoneadditionalnodeuiontoit,whichwecallG1andV1=V[UwhereU=fuig.ThenaccordingtoLemma 27 ,apower-lawgraphG0=(V0,E0)canbeconstructedbyembeddingG1andabipartitegraphG2=(V12,V22;E2)whereV12,V22aretwosetsofdisjointnodesinG2andmaxf4,log(2n)+log(2n+1)gwithsomespecic.NotethatV12andV22aremarkedgrayandwhiteseparatelyasshowninFig. 4-3 .WeshowthatthereisaVCofsizekinGiffG0hasaCNDS0ofsizek0suchthatthepairwiseconnectivityofG0[V0nS0]isatmostc,wherek0=k+minfjV12j,jV22jgandc=n)]TJ /F7 11.955 Tf 11.96 0 Td[(k.First,supposeS2VisavertexcoverofGwithjSj=k.Therefore,GhasavertexcoverSofsizekiffG[G2hasavertexcoverS0ofsizek+minfjV12j,jV22jgsinceVCispolynomiallysolvableinanybipartitegraphsaccordingtoKonig0sTheorem[ 49 ].Then,afterremovingS0fromG0,weonlyhavealldisjointlinks(vi,ui)leftwherevi62S.Therefore,thepairwiseconnectivityonpower-lawgraphG0isn)]TJ /F7 11.955 Tf 11.96 0 Td[(k,whichisequaltoc. 88

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AAninstance BReducedgraphG0=G1[G2Figure4-3. AnexampleofCNDreductiononPLGs.Forsimplicity,wejustdrawthenodesinGanditsnewlyaddednodesandlinks. Conversely,supposethatS0V0withjS0j=k0isaCNDofG0,thatis,thetotalpairwiseconnectivityofG0[V0nS0]isatmostc.First,ifui2S0,itiseasytoverifythatreplacinguiwithanyviwillfurtherdecreasethepairwiseconnectivity.SincejS0j=k0=k+minfjV12j,jV22jg,wecaneasilymodifyS0tobeavertexcoverofG[G2,wherethetotalpairwiseconnectivityonG0isatmostc=n)]TJ /F7 11.955 Tf 12.16 0 Td[(k.ThusS0\VisaVCofG. 4.2.2HILPRApproachApartfromtheabovetheoreticalhardnessresultsforCLDandCND,thesetwoproblemsareusuallyevenhardertobeapproximated.ThepairwiseconnectivitycaneitherremainO(n2)forCLDindensenetworksevenwhenkislarge,orreach0forCNDwhenkislargerthanthesizeofvertexcover.Inthissection,wepresentoursolution,aHybridIterativeLinearProgrammingRounding(HILPR)algorithmtobothCLDandCNDproblems.Inabigpicture,HILPRformulatesCLDandCNDunderIntegerLinearProgramming(ILP)formulations,andthensolvesthemusinganiterativeroundingtechnique.Inaddition,HILPRalsotakesintoaccountthelocalsearchand 89

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constraintpruningtechniquesinordertofurtherimproveitsefciencyandreduceitstimecomplexity. 4.2.2.1IntegerlinearprogrammingformulationCriticalLinkDisruptorForeachpairofnodesi,j2V,wedeneanindicatorvariableuijas:uij=8>><>>:1,ifiandjareconnected0,otherwiseThenwehavethefollowingILP: minXi,j2Vuijs.t.uij+ujh)]TJ /F7 11.955 Tf 11.95 0 Td[(uhi18i,j,h2VX(i,j)2Ewij(1)]TJ /F7 11.955 Tf 11.95 0 Td[(uij)kuij2f0,1g (4) wheretheobjectiveistominimizethetotalpairwiseconnectivity.Therstconstraintimposesthetriangularconnectivity.Thatis,ifnodeiandjareconnected,nodejandhareconnected,nodeiandhhavetobeconnected.Thesecondconstraintmeansthatthetotalweightofalldeletedlinkshastobeatmostk.Wenotethatforanedge(i,j)2E,ifuij=0intheILPsolution,thenthatlink(i,j)isacriticallink.CriticalNodeDisruptorForCND,wesimplyextendtheaboveIPformulationforCLDin( 4 )as minXi,j2Vuijs.t.vi+vj+uij18(i,j)2Euij+ujh)]TJ /F7 11.955 Tf 11.96 0 Td[(uhi18i,j,h2VXi2Vwivikvi2f0,1g,uij2f0,1g (4) 90

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whereviisfurtherdenedasvi=8>><>>:1,ifnodeiisdeleted(i.e.,criticalnodes)0,otherwiseTherstadditionalconstraintguaranteesthatatleastoneendpointofalinkhastobedeletedifitstwoendpointsaredisconnectedintheoptimalsolution.OtherconstraintsarecarriedoutasCLD.WefurtherconstrainkinCNDtosatisfyLemma 29 forunweightedgraphstoavoidthezeropairwiseconnectivity,thatis,allnodesinnetworkareindependent. Lemma29. ForanunweightedgraphG,theoptimalpairwiseconnectivityofCNDislargerthan0ifk
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suchthatPvi2Ywi.Inthenextiteration,thegraphwillrstbeupdatedaccordingtothepreviousroundingresults,i.e.,theidentiedcriticallinksornodeswillberemovedfromthegraph.ThenLPwillbereformulatedaccordingthenewresidualgraph.Thealgorithmterminateswhenthetotalweightofallidentiedcriticallinks(ornodes)reachestok. Algorithm7:HILPRforagiven Input:GraphG=(V,E),anintegerk,Output:Thesetofcriticallinks/nodesS 1S ;; 2 // IterativeLPRounding 3whilek>0do 4ifk
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connectivityfunctionofG[EnS].Iff(G,S0)
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uij+ujk)]TJ /F7 11.955 Tf 11.95 0 Td[(uik=1,wehaveuhiuij+ujh)]TJ /F5 11.955 Tf 11.96 0 Td[(1uij+ujk+ukh)]TJ /F5 11.955 Tf 11.95 0 Td[(2=uik+ukh)]TJ /F5 11.955 Tf 11.95 0 Td[(1Thus,thetriangleinequalityofthetuple(i,k,h)issatised,shownasboldinFig. 4-4 .Oncethetriangleinequalityofthetuple(i,j,k)istight,thetriangleinequalityofthetuple(i,k,h)willbesatisedforallnodesh.Sincethenumberoftriangleinequalityconstraintsis3)]TJ /F8 7.97 Tf 5.48 -4.38 Td[(n3=O(n3),thenumberofactiveconstraintswillbeO(n3)=n=O(n2)afterpruningprocess. 4.2.2.3PerformanceevaluationPerformanceoftheHILRPAlgorithmThethreenetworksweusetoevaluatetheperformanceofourproposedHILPRalgorithmaredescribedasfollows: 1. TherealterroristnetworkcompiledbyKrebs[ 57 ]with62nodesand153links,whichreectstherelationshipbetweentheterroristsinvolvedintheterrorismattacksofSep.11,2001.ThisexperimentattemptstoevaluatetheperformanceofHILPRonareal-worldsocialnetwork.Inordertobreakdowntheterroristnetwork,wecancapturetheindividualscorrespondingtothecriticalnodesidentiedbyHILPR. 2. Waxmannetworktopology,awidely-acceptedInternetAStopologicalmodel,isgeneratedbythewell-knownBRITE[ 64 ]. 3. Power-lawnetworktopology,generatedbyBarabasigraphgenerator[ 1 ],hasbeendiscoveredasoneofthemostremarkablepropertiesinmanylarge-scalenetworkssuchastheInternetandthesocialnetworks.Tokeepthesimilardensityastherealterroristnetworkandalsoshowthecomparisonwithoptimalsolutions,weusetheinstancewith70nodesand140links.Wegenerate100instancesforbothWaxmanandpower-lawmodelsandshowtheaverageresults.InordertoshowtheeffectivenessofourproposedHILPRalgorithm,wecompareitwiththeoptimalsolutionobtainedbysolvingtheILPdirectly.WealsocompareHILPRwithtwocentralityapproaches:degreecentrality(DC)andbetweennesscentrality(BC),whichareoftenusedinnetworkanalysis[ 17 ].InDC,theklinksandnodesoflargest 94

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degreesareselectedascriticallinksandnodes,wherethedegreeofalink(u,v)isdenedasd(u)+d(v).Similarly,inBC,theklinksandnodeswithlargestbetweennessareselectedascriticallinksandnodes,wherethebetweennessofalinkoranodeisdenedasthenumberofshortestpathsamongallpairsofnodesthatpassesthroughit.ForCND,wefurthercompareHILPRwithCNLSapproachproposedbyArulselvanetal.[ 9 ],whichalsoaimstominimizethepairwiseconnectivity.Astheonlyfreeparameterinouralgorithm,werstcomparetheimpactsofdifferentvaluesinourexperimentssuchthatwecanbalancethesolutionqualityandrunningtimebycarefullyselectingthisexperimentalvalue.AsillustratedinFig. 4-5 ,theresultsreturnedbyouralgorithmareveryclosesolutions.Thus,weuse=1forCNDduetoitsslightlybetterperformance,and=5forCLDtoreducetherunningtimesincethenumberofcriticallinksisusuallylargercomparedwithcriticalnodes.Next,weshowthatourHILPRapproachreturnsavery-nearoptimalsolutionandoutperformsotherapproaches. ACriticalLinks BCriticalNodesFigure4-5. TheperformanceofHILPRusingdifferentinterroristnetwork Fig. 4-6 andFig. 4-7 reportthecomparisonoftheaboveHILPRalgorithmandcentralityalgorithmsforCLDandCNDontheabovethreedifferentnetworks.Inthesegures,wenoticethatthesolutionofHILPRalgorithmisverycloselyapproachingtheoptimalsolutionforbothCLDandCNDonallthesethreenetworks(NotethataportionofoptimalsolutionsofCNDinWaxmannetworksaremissingduetoits 95

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AWaxmannetworks BPower-lawnetworks CTerroristnetworkFigure4-6. TheperformanceevaluationofHILPRagainstthedegreeandbetweennesscentralityalgorithmsfortheCLDproblem AWaxmannetworks BPower-lawnetworks CTerroristnetworkFigure4-7. TheperformanceevaluationofHILPRagainstthedegreeandbetweennesscentrality,andCNLSalgorithmsfortheCNDproblem extremelyhighcomputationalcomplexity,whichisbecausethenetworkisneitheralmostintactnoralmostfragmented).ThepairwiseconnectivityderivedfromdegreecentralityalgorithmsismuchworsethanHILPRalgorithmbecausethelinksornodesofhigherdegreescouldalreadyconnectothercriticallinksornodesandthereforearenotnecessarytobecountedascriticalanymore.Forinstance,thehubnodes(nodesofhighdegree)inpower-lawnetworksarenotnecessarilyconnectedwitheachothersuchthattheremovaloftwohubnodescouldbelesseffectivetoreducethepairwiseconnectivitythantheremovaloftwoothernodeswhichcandisconnectthenetwork.Thebetweennesscentralityperformsworstduetothelackofallpathsinformationratherthanonlyshortestpaths.Thatis,apairofnodescanstillbeconnectedevenwhenonlytheshortestpathbetweenthemisdestroyed.ThereasonwhyourHILPRalgorithmoutperformsCNLSismainlybecauseofthedifferentstrategytochoosetheinitialcriticalnodesbeforedoingthelocalsearch.TheCNLSmethodisonlybasedonthemaximum 96

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degreefromamaximalindependentset,hopingthattheremovalofthesenodescangreatlyfragmentthenetwork.However,thisisnotalwaystruesincemanynodesinthemaximalindependentsetareusuallyoflowdegree,andconsequentlydonotplayanimportantroleindestroyingthenetwork.InourHILPRapproach,bysolvingtheLPandroundingthetopelementsiteratively,wetakeintoaccountallpossiblepathsandconnectionsbetweendifferentnodessuchthatthecriticalelementscanbeaccuratelyidentied. Figure4-8. OverlappingcriticalnodesbetweenoptimalsolutionandHILPRinterroristnetwork Specically,inordertofurthershowtheeffectivenessofourmetricandalgorithm,i.e.,thecriticallinksandnodesinreal-worldnetworkscanbecorrectlydetectedusingouralgorithm,wedigintotherealterroristnetworkinwhichtheidentitiesofnodesareavailable.TheresultsreturnedbyourHILPRalgorithmshowthatwecandetecttherealimportantpersonnelbyminimizingtotalpairwiseconnectivity.Forinstance,thetwonodes37and48intheterroristnetwork,whichhavebeenshownastheleadersin[ 57 ?],canbecorrectlydetectedusingourHILPRalgorithmaslongask=2.Yet,ifweusedegreeandbetweennesscentralitymethods,onlynode37canbedetectedwhenk=2andnode48willnotbedetecteduntilkischosentobe6and5respectively. 97

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Eventhoughourobjectiveistominimizethepairwiseconnectivity,wearestillinterestedintheoverlappingpercentageofcriticalnodesourHILPRalgorithmreturnsandtheoptimalcriticalnodes.AsreportedinFig. 4-8 ,intherealterroristnetwork,optimalcriticalnodescanbe100%successfullydetectedusingourHILPRalgorithminmorethan1/3casesfordifferentkvalues.Theaverageoverlappingpercentageisaround80%sincethereexistsomenodesplayingthesameroleinnetworkconnectivitysuchthatthepairwiseconnectivitystillcanbeminimizedalthoughourHILPRalgorithmidentiesdifferentcriticalnodesfromtheoptimalsolution.Moreover,therunningtimeofourHILPRalgorithmislessthan5secondsinallthesethreenetworks,fordetectingeithercriticallinksornodes,whichisonlyslightlyworsethancentralityalgorithms(1-2seconds)andCNLSalgorithm(2-3seconds).Especiallywhenkissmall,i.e.,onlythemostcriticalelementsarerequiredtobedetected,ouralgorithmcannisharound3seconds,whichfurtherillustratestheeffectivenessofourHILPRalgorithmintermsofbothsolutionqualityandrunningtime.MetricEvaluationWeevaluatetheresidualnetworkobtainedbyHILPRalgorithmundervariousnetworkvulnerabilitymetrics.AshasbeenshowninFig. 4-6 and 4-7 ,degreecentralityandbetweennesscentralitycannotaccuratelyreectthenetworkvulnerability.Therefore,wefocusonthefollowingthreeothermetrics:(1)averageshortestpathlength(ASP)betweeneachnode-pairs(theshortestdistanceis0ifthepairofnodesarenotconnected),(2)averageavailableows(AAF)betweeneachnode-pairs,and(3)globalclusteringcoefcients(GCC)denedas#closedtriplets #connectedtriplesofvertices,inwhichaclosedtripletconsistsofthreenodesthatareconnectedbythreeundirectedties.Particularly,weareinterestedtoseehowthevaluesofthesemetricschangeintheresidualnetworkafterweremovethecriticalelementswhichcansuccessfullyreducethepairwiseconnectivityofthenetwork.SinceourHILPRalgorithmcansuccessfullydetecttherealcriticallinksandnodesasdiscussedintheprevioussubsection,we 98

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condentlyevaluatetheabovethreemetricsontheresidualgraphsobtainedbyHILPRalgorithm.Fig. 4-9 showsthechangesinvaluesoftheabovethreemetricsafterremovingdifferentnumberofcriticallinksornodes.Unfortunately,noneofthesethreemetricsinresidualnetworkscanclearlycastthenetworkvulnerability.AsfortheASP,wecanonlyconsidertheASPwithineachconnectedcomponentafterthenetworkisdisconnected;otherwisetheASPbecomesinniteandthereforefailstomeasurethenetworkvulnerability.However,thevalueofASPwithinconnectedcomponentsiseitherirregular(Fig. 4-9A )orcontrarytotheintuition,i.e.,ASPusuallyincreaseswithafterremovingcriticalelements(Fig. 4-9B ).Similarly,theAAFfailstoassessthenetworkvulnerabilityduetoitsirregularityforcriticallinks.ThemonotonousdeceaseofAAFintheresidualnetworksafterremovingcriticalnodesisgreatlyduetothedisconnectionofthenetwork,whichreducestheowfromtwonodesindifferentconnectedcomponentsto0.However,thenodesdisconnectingthenetworkarenotnecessarytobecriticalnodes.Atlast,thevariationofGCCvaluesisirregularforbothcriticallinksandnodesduetothesimultaneousdecreaseofthenumberofconnectedtriplesofvertices.Particularly,whenthenetworkishighlyfragmented,thismetriccaneasilybecomeinniteandmeaningless(intheresidualnetworkofk22asshowninFig. 4-9B )sincethenumberofconnectedtriplesofverticesbecomes0. ACriticalLinks BCriticalNodesFigure4-9. Thecomparisonofdifferentmetricsonterroristnetwork 99

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4.2.3TRGAApproachunderCascadingFailuresAsillustratedinSection 3.7 ,thecascadingfailurescouldleadtotheentiredifferentsetofcriticalelements.Inthissubsection,weproposeoursolution,aTracebackandLPRounding-GreedyAlgorithm(TRGA)forsolvingtheCCNDproblem. 4.2.3.1TRGA:aniterative2-phasealgorithmInabigpicture,TRGAalgorithm(Algorithm 9 )iterativelydetectsthemostvulnerablenodesuntilkmostvulnerablenodes,inwhicheachiterationistwo-fold:(1)identifyingtheultimatefailurenodesaftercascadingfailures;(2)tracingbackthevulnerablenodesbasedontheabovefailurenodes.Inaddition,TRGAalsotakesintoaccountthelazy-updateandconstraintpruningtechniquesineachiterationfurtherreduceitstimecomplexity.Intheend,alocalsearchisprovidedtoimprovethesolutionquality.Therestofthissubsectiondiscussestwostepsineachiterationindetail.Phase1:UltimateFailureNodesIdenticationInordertodetecttheultimatefailurenodes,theideaistorstguesstheextentoffragmentationinresidualnetworks,i.e.,thenumberofconnectednode-pairsatlast,andthenidentifythesenodesbasedontheiterativeroundingapproachin[ 77 ],whichhasbeenshowtobeoneofthebestapproachesfordetectingcriticalnodeswhenfailuresarenotcascaded.DenotingtheresidualpairwiseconnectivityasP,weestimateitbasedonthefollowingintuitionandobservation:thelargerthedegree,themorevulnerablethenodeisinanetwork[ 84 ].Therefore,ineachiteration,wechoosethek0highestdegreesintheresidualnetwork(k0=k)]TJ /F1 11.955 Tf 12.62 0 Td[(#detectedvulnerablenodes)tosimulatethecascadingfailuresafterdeletingthemandobtainthepairwiseconnectivityP.Then,wehavethefollowingIntegerLinear 100

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Programming(ILP)formulation: minXi2Vvis.t.vi+vj+uij18(i,j)2Euij+ujh)]TJ /F7 11.955 Tf 11.96 0 Td[(uhi18i,j,h2VXi,j2VuijPvi2f0,1g,uij2f0,1g (4) whereviisfurtherdenedasvi=8>><>>:1,ifnodeiisdeleted(i.e.,vulnerablenodes)0,otherwiseanduij=8>><>>:1,ifiandjareconnected0,otherwiseTherstadditionalconstraintguaranteesthatatleastoneendpointofalinkhastobedeletedifitstwoendpointsaredisconnectedintheoptimalsolution.ThesecondconstraintimposesthetriangularconnectivitywhilethethirdconstraintmeansthatthetotalpairwiseconnectivityaftercascadingfailureshastobeatmostP.Tosolveiteffectively,weborrowtheideain[ 77 ]torelaxtheequation( 4 ),iterativelysolvetheLPandroundthelargestvi.Likewise,wefurtherapplyconstraintspruninginsolvingtheLPandlocalsearchattheendofthewholeultimatefailurenodesidentication.Phase2:VulnerableNodesTracingBackWiththesetofultimatefailurenodesaftercascadingfailures,wetracebacktothevulnerablenodesbasedonthefollowinggreedyalgorithm.Inparticular,ineachiteration,weselectanodewhichcanleadtothecollapseofmostultimatefailurenodes,callcascadinginuence,bysimulatingthefailurecascadesafterremovingeachnode. 101

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Toobtainthecascadinginuenceofeachnodeateachiteration,theeasiestapproachistorecomputeforeachnode,yetthisapproachisextremelytime-consuming.Instead,weapplythelazy-updateprocessaftertheinitialfailureinuence.Specically,afterthesimulationattherstiteration,wemaintainamaxpriorityqueueQinwhichthepriorityistheircascadinginuence.Ineachiteration,thenodeuwiththehighestcascadinginuenceisextractedandwerecomputetheextranodesneededtofailu.Inthenextiteration,uwillbeselectedifitstillhasthehighestpriority.Otherwise,uispushedbacktothepriorityqueue,meanwhilethenewnodewiththehighestprioritywillbepicked.Notethatifthenumberofselectedvulnerablenodesarelargerthank0,wewillchoosethek0highestdegreesintheresidualnetworkfrompreviousroundinsteadandmoveontothelocalsearchphasedirectly. 4.2.3.2OptimalityofCCNDproblemInthissubsection,weproposethefollowingIntegerLinearProgramming(ILP)formulationforCCNDprobleminordertoobtainitsoptimalsolution.Next,weapplyasparsemetrictechniquetofurtherreducethenumberofconstraints,meanwhilekeepthesameoptimalresult.MathematicalFormulationForeachpairofnodesi,j2V,wedeneanindicatorvariableuijas:uij=8>><>>:1,ifiandjareconnected0,otherwiseandforallintegerst2[0,d],wedenevti=8>><>>:1,ifnodeifailsinroundt0,otherwise 102

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Notethatv0i=1whennodeiisavulnerablenodeandfailsatthebeginning.ThenwehavethefollowingILP: minXi,j2Vuijs.t.vdi+vdj+uij18(i,j)2Euij+ujh)]TJ /F7 11.955 Tf 11.96 0 Td[(uhi18i,j,h2VXi2Vv0ikXj2N(vi)vt)]TJ /F9 7.97 Tf 6.58 0 Td[(1j+deg(vi)vt)]TJ /F9 7.97 Tf 6.58 0 Td[(1ideg(vi)vti8i2V,80tdvtivt)]TJ /F9 7.97 Tf 6.58 0 Td[(1i80td8i2V,0tdvti2f0,1g80tduij2f0,1g (4) wheretheobjectiveistominimizethetotalpairwiseconnectivity.Therstconstraintguaranteesthatatleastoneendpointofalinkhastobedeletedafterdroundcascadesifitstwoendpointsaredisconnectedintheoptimalsolution.Thesecondconstraintimposesthetriangularconnectivity.Thatis,ifnodeiandjareconnected,nodejandhareconnected,nodeiandhhavetobeconnected.Thethirdconstraintmeansthatthetotalpairwiseconnectivityafterdroundfailurecascadesisatmostfractionofallnode-pairs.Thelasttwoconstraintsdealswiththecascadesprocessandkeepsfailednodestobefailureinthefollowingroundsrespectively. 4.2.3.3ExperimentalevaluationInthissection,weevaluatetheperformanceofourTRGAalgorithmondifferenttypesofsyntheticandreal-worldnetworks.ThesimulationisimplementedusingtheCPLEXoptimizationsuitefromILOG,whichincludesthesimplexmethod[ 42 ],thebranch&boundalgorithm,andadvancedcutting-planetechniques[ 86 ]. 103

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ThethreenetworksweusetoevaluatetheperformanceofourproposedTRGAalgorithmaredescribedasfollows:(1)USNetworkAssetscompiledby[ 77 ]with71nodesand98edges,whichprovidesthecurrentcustomerneedsinXOCommunicationsservice.ThisexperimentattemptstoevaluatetheperformanceofTRGAonareal-worldcommunicationnetwork.Inordertomaintainthefunctionalityofthiscommunicationnetwork,weneedtoprotectthemostcriticalISPscorrespondingtothevulnerablenodesidentiedbyTRGA;(2)Power-lawnetworktopologygeneratedbyigraphlibrary[ 26 ]usingthemodelin[ 2 ],with=1.8and70nodes;(3)Small-worldnetworktopologygeneratedbyigraphlibrary[ 26 ]usingWattsandStrogatzmodelin[ 70 ],withk=2,=0.2and70nodes.TheselectionofparametersinthesetwosyntheticnetworksistokeepthesimilardensityastheUSNetworkAssetsnetworkandalsoshowthecomparisonwithoptimalsolutions.Wegenerate100instancesforbothpower-lawandsmall-worldnetworksandshowtheaverageresults.InordertoshowtheeffectivenessofourproposedTRGAalgorithm,wecompareitwiththeoptimalsolutionobtainedbysolvingtheILP( 4 )directly.WealsocompareTRGAwithtwocentralityapproaches:degreecentrality(DC)andbetweennesscentrality(BC),whichareoftenusedinnetworkanalysis[ 17 ].InDC,theknodesoflargestdegreesareselectedasvulnerablenodes,andinBC,thekandnodeswithlargestbetweennessareselectedasvulnerablenodesobtainedusing[ 19 ],wherethebetweennessofanodeisdenedasthenumberofshortestpathsamongallpairsofnodesthatpassesthroughit.Fig. 4-10 reportsthecomparisonoftheaboveTRGAalgorithmandcentralityalgorithmsforCCNDontheabovethreedifferentnetworks.Inthesegures,wenoticethatthesolutionofTRGAalgorithmisverycloselyapproachingtheoptimalsolutionforbothCCNDonallthesethreenetworks.Exceptinpower-lawnetworksinwhichnodesofhighdegreeshavebeenshownasimportantnodes[ 84 ],thepairwiseconnectivityderivedfromdegreecentralityalgorithmsismuchworsethanTRGAalgorithmespecially 104

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AUSNetworkAssets BPower-LawNetworks CSmall-WorldNetworksFigure4-10. TheperformanceevaluationofTRGAagainstdegreeandbetweennesscentralityalgorithmsfortheCCNDproblem insmall-worldnetworksduetotheirhomogeneityinnodedegreessuchthatthenodesofhigherdegreescouldalreadyconnectothervulnerablenodesandthereforearenotnecessarytobecountedasvulnerableanymore.Thebetweennesscentralityperformsworstinbothpower-lawnetworksandUSNetworkAssetsduetothelackofallpathsinformationratherthanonlyshortestpaths.Thatis,apairofnodescanstillbeconnectedevenwhenonlytheshortestpathbetweenthemisdestroyed.Yet,itoutperformsdegreecentralityinsmall-worldnetworks,inwhichthedifferenceofdegreesisnotsubstantial.InourTRGAapproach,intherstphaseofeachiteration,bysolvingtheLPandroundingthetopelementsiteratively,wetakeintoaccountallpossiblepathsandconnectionsbetweendifferentnodessuchthatthecriticalelementscanbeaccuratelyidentied.Meanwhile,thesecondphaseinTRGAwiththe 105

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back-tracingcanalsopreciselydetecttheoriginalvulnerablenodesbyprovidingmoreinformationthanonlydegreeorbetweenness.Moreover,therunningtimeofourTRGAalgorithmislessthan10seconds(duetotheLPsolver)inallthesethreenetworks,fordetectingvulnerablenodes,whichisacceptablecomparedwithcentralityalgorithms(1-2seconds),andover100timesfasterthanobtainingoptimalsolutionevenwithsparsemetric.Besides,USNetworksAssetsasawell-designedcommunicationnetworkinpractice,evenwithlowerdensity,isshowntobethemostrobustamongthesethreetopologies. 4.3RelatedWorksManyexistingworksonnetworkvulnerabilityassessmentmainlyfocusonthecentralitymeasurements[ 17 ],includingdegree,betweennessandclosenesscentralities,averageshortestpathlength[ 3 ],globalclusteringcoefcients[ 60 ].Duetothefailurestoassessthenetworkvulnerabilityusingabovemeasurements,Sunetal.[ 80 ]rstproposedthetotalpairwiseconnectivityasaneffectivemeasurementandempiricallyevaluatethevulnerabilityofwirelessmultihopnetworksusingthismetric.Arulselvanetal.[ 9 ]showedthechallengeofCNDproblembyprovingitsNP-completeness.Lateron,the-disruptorproblemwasdenedbyDinhetal.[ 27 ]tondaminimumsetoflinksornodeswhoseremovaldegradesthetotalpairwiseconnectivitytoadesireddegree.TheyprovedtheNP-completenessofthisproblemwithrespecttobothlinksandnodesandthecorrespondinginapproximabilityresults.Evenforthetreetopology,DiSummaetal.[ 61 ]foundthatthediscoveryofcriticalnodesalsoremainsNP-completeusingthismetric.Inthispaper,wefurtherinvestigatethetheoreticalhardnessofbothCLDandCNDonUDGsandPLGs.Inaddition,thereareafeweffectivesolutionsintheliteratureofthenetworkvulnerabilityassessmentbasedonthepairwiseconnectivity.Arulselvanetal.[ 9 ]designedaheuristic(CNLS)todetectcriticalnodes,whichishoweverstillfarawayfromtheoptimalsolutioninlarge-scaleanddensenetworks.In[ 27 ],Dinhetal.proposed 106

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pseudo-approximationalgorithmstosolvethe-disruptorproblem.However,thisproblemisdeneddifferentlythanoursandhardtouseitssolutionwhenweonlyknowtheavailablecosttodestroyorprotectthesecriticallinksornodes.Whenfailuresarecascaded,theseresultsarenolongervalid,inwhichthevulnerabilityofnetworkscouldbesubstantiallydifferent.Mostoftheworksregardingcascadingfailuresmainlyfocusonmodels[ 25 45 85 ].Moreover,therearesomeotherpapersprovidingsomeexperimentalanalysis[ 29 67 ].Unfortunately,thetheoreticalworksarelacked,whicharecrucialtothenetworkdesignandproactiveprotection.Therefore,weprovideaprobabilisticanalysistoassessthevulnerabilityforcomplexnetworksinthecaseofcascadingfailures,leadingtodeepinsightstotherobustnessofvariousnetworksunderrandomfailures.Inaddition,mostofworksonnetworkvulnerabilityassessmentforadversarialattacksarealsostudiedwithouttakingintoaccountthecascadingfailures.Besidesthewidely-usedcentralitymeasurements[ 3 17 60 ],Arulselvanetal.[ 9 ]rstproposedthetotalpairwiseconnectivityasaneffectivemeasurement,basedonwhichtheyproposetheCNDproblemanddesignedaheuristictodetectcriticalnodes.The-disruptorproblemwaslaterdenedbyDinhetal.[ 27 ]followedbypseudo-approximationalgorithms.Unfortunately,theseapproachesfailtoaccuratelyidentifythevulnerablenodesinthepresenceofcascadingfailures.Inthispaper,wefurtherinvestigatethetheoreticalhardnesstheCVNDproblem,alongwithaneffectivealgorithm. 107

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Algorithm9:TRGAAlgorithm Input:NetworkG,ThresholdOutput:ThesetofkvulnerablenodesS 1k0 k; 2S ;; 3whilejSjPdo 10UseConstraintPruningin[ 77 ]tosolvetheLPformulationwithP; 11P P)]TJ /F1 11.955 Tf 12.62 0 Td[(disconnectednode-pairsafterremovingu; 12u thenodewithlargestvi; 13U U[fug; 14G G[Vnfug]; 15end 16 // VulnerableNodesTracingBack 17Q ;; // PriorityQueue 18S0 ;; 19while9onenodedoesnotfaildo 20ifQ=;then 21foreachnodeudo 22CalculatethecascadinginuenceafterremovingufromG; 23end 24ConstructQbasedoncascadinginuenceofeachnode; 25end 26else 27S0 S0[thenodeinQwithmaxpriority; 28Updatecascadinginuencecausedbyremovingthisnode; 29end 30end 31ifjS0j>k0then 32S S[k0largestdegreenodesinG[VnS]; 33end 34else 35S S[S0; 36end 37end 38 // LocalSearch 39S S; 40foreachnodeu2Sdo 41Swapping(u);(Algorithm2in[ 77 ]byreplacingf(G,S0)withthepairwiseconnectivityfunctionofresidualgraphGafterremovingS0); 42end 43S S; 44returnS; 108

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CHAPTER5CONCLUSIONInthisdissertation,werstanalyzedtheapproximationhardnessandinapproximabilityofoptimalsubstructureproblemsonpower-lawgraphs.Theseproblemsareonlyillustratedintheliteraturenotbeabletoapproximatedintosomeconstantfactorsonbothgeneralandsimplepower-lawgraphsalthoughtheyremainAPX-hard.Onthecontrary,wealsoshowthatMaxCliqueandGraphColoringarestillveryhardtobeapproximatedsincetheoptimalsolutionstotheseproblemsaredependentonthestructureoflocalgraphcomponentratherthanglobalgraph.Inotherwords,thepower-lawdistributionindegreesequencedoesnothelpmuchforsuchoptimizationproblemswithoutthepropertyofoptimalsubstructure.Moreover,weproposedaalgorithmframework,alongwithatheoreticalframeworkforanalyzingapproximationratios,basedontheideaofpercolatingthepower-lawgraphfromthenodesoflowestdegreetoothernodes.Inaddition,westudytherobustnessofpower-lawnetworksundervariousthreats,i.e.randomfailures,preferentialattacksanddegree-centralityattacks.Essentially,thepower-lawnetworksareillustratedtoextremelytoleraterandomfailures.Inthemeanwhile,theyaremorerobustunderbothpreferentialattacksanddegree-centralityattacksiftheyhaveasmallerexponentialfactor.Whenfailurescanbecascaded,weshowedthatpower-lawnetworksareextremelyvulnerableevenwithverysmall.Inordertoprovideanoptimaldesignofpower-lawnetworks,wefurtherexploitthetopologiesofpracticalreal-worldnetworksbyoptimizingthecostsandguaranteeingtheirrobustness.Thebestrangeoftheexponentialfactorisillustratedtobe[1.8,2.5],whichgivesareasonableexplanationforthetopologiesofmostreal-worldnetworks.When<1.8,thenetworkmaintenancecostisveryexpensive,andwhen>2.5,thenetworkrobustnessisunpredictablesinceitdependsonthespecicattackingstrategy.Also,westudyCLDandCNDoptimizationproblemstoidentifycriticallinksandnodesin 109

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anetworkwhoseremovalsmaximallydestroythenetwork'sfunctions.WeprovedtheirNP-hardnessandproposedHILPR,anovelLP-basedroundingalgorithm,forefcientlysolvingCLDandCNDproblemsinatimelymanner.Inthepresentofcascadingfailures,wefurtherstudyCCNDproblemanddevelopedtheeffectiveiterative2-phaseTRPAalgorithm.Theexperimentsonvarioussyntheticandreal-worldnetworksillustratedthegoodperformanceofourproposedapproaches. 110

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BIOGRAPHICALSKETCH YilinShenreceivedhisPh.D.fromtheUniversityofFloridainthespringof2013andhisB.S.degreeinappliedmathematicsfromDonghuaUniversity,Shanghai,China,in2005.Hisresearchfocusesonvulnerabilityassessmentandsecurityofcomplexnetworks,includingcommunicationnetworks,wirelesssensornetworksandsocialnetworks,anddesigningapproximationalgorithmsfornetworkoptimizationproblems.HeisastudentmemberoftheIEEE. 119