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Cascading Propagation and Optimization in Networks

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Title:
Cascading Propagation and Optimization in Networks
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1 online resource (103 p.)
Language:
english
Creator:
Nguyen, Dung T
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
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Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Computer Engineering, Computer and Information Science and Engineering
Committee Chair:
Thai, My Tra
Committee Members:
Entezari, Alireza
Kahveci, Tamer
Sahni, Sartaj Kumar
Smith, Jonathan Cole

Subjects

Subjects / Keywords:
cascading -- centrality -- coupling -- diffusion -- interdependent -- optimization
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

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Abstract:
Cascading processes are more and more popular in highly connected networks. These processes are recognized in a wide range of networks with different contexts: the information diffusion in online social networks, the cascading crisis in the network of banks, the cascading failure in power networks, etc. Regardless of the network type and mechanism, they still share fundamental properties: (1) the root cause is the influences/dependencies between nodes of one or more networks, (2) the process often starts from a small group of nodes, (3) the impact is high due to the large number of involved nodes. It is thus crucial to study these processes and exploit them efficiently.  In this work, we study several optimization problems relating to the cascading process in networks. In particular, we mainly focus on two kinds of problems: (1) finding a small set of nodes which can maximize the impact through the cascading process and (2) finding a set of nodes with minimum size which causes the desired impact. Depending on the cascading mechanism, we design different strategies to solve the problem efficiently by exploiting both the properties of both the cascading mechanism and the network structure.
General Note:
In the series University of Florida Digital Collections.
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Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Dung T Nguyen.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Thai, My Tra.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-08-31

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lcc - LD1780 2013
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UFE0045910:00001

MISSING IMAGE

Material Information

Title:
Cascading Propagation and Optimization in Networks
Physical Description:
1 online resource (103 p.)
Language:
english
Creator:
Nguyen, Dung T
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Computer Engineering, Computer and Information Science and Engineering
Committee Chair:
Thai, My Tra
Committee Members:
Entezari, Alireza
Kahveci, Tamer
Sahni, Sartaj Kumar
Smith, Jonathan Cole

Subjects

Subjects / Keywords:
cascading -- centrality -- coupling -- diffusion -- interdependent -- optimization
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:
Cascading processes are more and more popular in highly connected networks. These processes are recognized in a wide range of networks with different contexts: the information diffusion in online social networks, the cascading crisis in the network of banks, the cascading failure in power networks, etc. Regardless of the network type and mechanism, they still share fundamental properties: (1) the root cause is the influences/dependencies between nodes of one or more networks, (2) the process often starts from a small group of nodes, (3) the impact is high due to the large number of involved nodes. It is thus crucial to study these processes and exploit them efficiently.  In this work, we study several optimization problems relating to the cascading process in networks. In particular, we mainly focus on two kinds of problems: (1) finding a small set of nodes which can maximize the impact through the cascading process and (2) finding a set of nodes with minimum size which causes the desired impact. Depending on the cascading mechanism, we design different strategies to solve the problem efficiently by exploiting both the properties of both the cascading mechanism and the network structure.
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 Dung T Nguyen.
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 2015-08-31

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045910:00001


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CASCADINGPROPAGATIONANDOPTIMIZATIONINNETWORKSByDUNGT.NGUYENADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013DungT.Nguyen 2

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IdedicatethistotheinternalGod 3

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ACKNOWLEDGMENTS IwouldliketothankmyadvisorProf.MyT.ThaiforgreatadvicesduringthePhDtime.Iwasbestowedapreciousgiftworkingwithherinfouryears.Hergoodexamplesaremytreasurewhichhavegraduallymademestrongerinbothdoingresearchandovercomingdifculties.ItwasluckyformetogetinfectedbyherpassionandcuriositywhenIwastryingtosolvechallengingproblems.WhenIwaslackingofmotivation,theritualtogetmotivatedmaybequitesimple:seeingherasanexample.IamthankfultoProf.AlirezaEntezari,Prof.TamerKahveci,Prof.SartajK.Sahni,andProf.J.ColeSmithfortheirtimeandconstructiveopinions.Iwouldliketothankallofmyclosedandacquaintedfriendsforsharingtheirknowledge,perspectives,opinions,andenjoyingmoments. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 11 1.1CascadingFailureinaNetwork ........................ 12 1.2CascadingFailureinInterdependentNetworks ............... 12 1.3InuenceDiffusioninMultipleOnlineSocialNetworks ........... 13 1.4Organization .................................. 14 2CASCADINGFAILUREUNDERLOADREDISTRIBUTIONINNETWORKS .. 15 2.1NetworkModelandProblemFormulation .................. 16 2.1.1GraphNotations ............................ 16 2.1.2CascadingFailureModel ........................ 16 2.1.3ProblemDenition ........................... 18 2.2InapproximabilityResult ............................ 18 2.3CascadingPotentialandDerivedAlgorithms ................ 22 2.3.1CascadingPotential .......................... 22 2.3.2CascadingPotentialAlgorithm ..................... 23 2.3.3AdaptiveCascadingPotentialAlgorithm ............... 24 2.3.4FullyAdaptiveCascadingPotentialAlgorithm ............ 26 2.4CooperatingAttackAlgorithm ......................... 27 2.5ExperimentalEvaluation ............................ 30 2.5.1Datasets ................................. 31 2.5.2TheperformanceofDifferentAlgorithms ............... 32 2.5.3NetworkRobustnessUnderDifferentSettings ............ 34 2.5.4VertexLoadandNetworkRobustness ................ 36 2.5.5NetworkTopologyandNetworkRobustness ............. 37 2.6RelatedWorks ................................. 37 2.7Summary .................................... 38 3CASCADINGFAILUREOFNODESININTERDEPENDENTNETWORKS ... 39 3.1NetworkModelandProblemDenition .................... 42 3.1.1InterdependentNetworkModel .................... 42 3.1.2CascadingFailuresModel ....................... 42 3.1.3ProblemDenition ........................... 42 5

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3.2ComputationalComplexity ........................... 43 3.3GreedyFrameworkforIPNDProblem .................... 45 3.3.1MaximumCascading(Max-Cas)Algorithm .............. 45 3.3.2IterativeInterdependentCentrality(IIC)Algorithm .......... 47 3.3.2.1Updatingfunction ...................... 48 3.3.2.2Convergence ......................... 49 3.3.3HybridAlgorithm ............................ 53 3.4ExperimentalEvaluation ............................ 54 3.4.1DatasetandMetric ........................... 54 3.4.2PerformanceofProposedAlgorithms ................. 55 3.4.3VulnerabilityAssessmentofInterdependentSystems ........ 57 3.4.3.1Differentcoupledcommunicationnetworks ........ 57 3.4.3.2Disruptorthreshold ..................... 58 3.4.3.3Differentcouplingschemes ................. 59 3.5RPDCC/RNDCCCouplingSchemes .................... 60 3.6RelatedWorks ................................. 62 3.7Summary .................................... 62 4INFLUENCEDIFFUSIONINMULTIPLEONLINESOCIALNETWORKS .... 64 4.1NetworkModelandProblemDenition .................... 67 4.1.1GraphNotations ............................ 67 4.1.2InuencePropagationModel ..................... 68 4.1.3ProblemDenition ........................... 69 4.2NetworkAlignment ............................... 69 4.3LosslessCouplingSchemes ......................... 71 4.3.1CliqueLosslessCouplingScheme .................. 72 4.3.2StarLosslessCouplingScheme .................... 77 4.4LossyCouplingSchemes ........................... 78 4.5InuenceRelay ................................. 82 4.6ExperimentalEvaluation ............................ 86 4.6.1Datasets ................................. 87 4.6.2ComparisonofCouplingSchemes .................. 88 4.6.3BenetsofCoupledNetwork ..................... 91 4.6.4BiasinSelectingSeedNodes ..................... 93 4.7ExtensionstoOtherCascadingModels ................... 95 4.8Summary .................................... 96 5CONCLUSIONS ................................... 97 REFERENCES ....................................... 98 BIOGRAPHICALSKETCH ................................ 103 6

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LISTOFTABLES Table page 4-1Foursquare-Twitterandco-authornetworkdata-sets ............... 87 7

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LISTOFFIGURES Figure page 2-1Whennodeufails,itsloadisredistributedtotheneighbornodes.Amongthesenode,vreceivesahighportionofloadfromuandbecomesoverloaded.Theloadofvisredistributedtoitsneighborswhichmakeszfail.Finally,theloadfromzcontinuestocausewfailandtheprocessstops. ............. 17 2-2ReductionfromMIN3SC2toCasCN ........................ 19 2-3VulnerabilityofWSNnetworkundernormalsetting ................ 33 2-4VulnerabilityofWSNnetworkundersafetysetting ................ 34 2-5VulnerabilityofWSNnetworkunderscaledsafetysetting ............ 35 2-6Networkrobustnesswithdifferentfailuretoleranceschemes ........... 36 2-7Networkrobustnessunderdifferentloaddistribution ............... 36 2-8Networktopologyandrobustness ......................... 37 3-1ExampleofInterdependentPowerNetworkandCommunicationNetwork ... 40 3-2AnexampleofreductionfromMIStoIPND .................... 44 3-3PerformanceComparisononDifferentInterdependentSystems:WSSystem(A,B),SSSystem(C,D),andEq-SSSystem(E,F). ............... 56 3-4TheVulnerabilityOfAFixedPowerNetwork .................... 58 3-5TheDisruptorThresholdwithDifferentNetworkSizes .............. 59 3-6VulnerabilityComparisonusingDifferentCouplingSchemes ........... 61 4-1Autoupdateacrosssocialnetworks ........................ 66 4-2ThenumberofsharedusersbetweenmajorOSNsin2009[ 2 ] ......... 66 4-3Anexampleoflosslesscouplingscheme ..................... 74 4-4StarSynchronization ................................. 78 4-5Lossycouplednetworkusingeasinessparameters.Thenumberofedgesismuchlessthanthelosslesscouplednetwork. ................... 80 4-6ComparingCouplingSchemesforFindingMinimumSeedSetonco-authorNetworks(uppergures)andonFSQandTwitter(lowergures) ........ 89 4-7Comparingcouplingschemeswithdifferentoverlappingfractionf ....... 90 4-8Comparingcouplingschemeswithdifferentnumberofpropagationhopsd .. 90 8

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4-9Thequalityofseedsetswithandwithoutusingthecouplednetwork ...... 92 4-10Thequalityofseedsetswithandwithoutusingthecouplednetwork ...... 93 4-11Thesupportbetweennetworksontheinuencepropagationofanetworkwithd=4(uppergures)andd=8(lowergures)hops.C,H,N,F,andTaretheabbreviationsofCM,Het,NetS,FSQ,andTwitter. .............. 94 4-12Thebiasinselectingseednodesonsynthesizednetworks(uppergures)andonFSQandTwitter(lowergures) ....................... 95 4-13Theinuencecontributionofseednodesfromcomponentnetworks ...... 96 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyCASCADINGPROPAGATIONANDOPTIMIZATIONINNETWORKSByDungT.NguyenAugust2013Chair:MyT.ThaiMajor:ComputerEngineeringCascadingprocessesaremoreandmorepopularinhighlyconnectednetworks.Theseprocessesarerecognizedinawiderangeofnetworkswithdifferentcontexts:theinformationdiffusioninonlinesocialnetworks,thecascadingcrisisinthenetworkofbanks,thecascadingfailureinpowernetworks,etc.Regardlessofthenetworktypeandmechanism,theystillsharefundamentalproperties:(1)therootcauseistheinuences/dependenciesbetweennodesofoneormorenetworks,(2)theprocessoftenstartsfromasmallgroupofnodes,(3)theimpactishighduetothelargenumberofinvolvednodes.Itisthuscrucialtostudytheseprocessesandexploitthemefciently.Inthiswork,westudyseveraloptimizationproblemsrelatingtothecascadingprocessinnetworks.Inparticular,wemainlyfocusontwokindsofproblems:(1)ndingasmallsetofnodeswhichcanmaximizetheimpactthroughthecascadingprocessand(2)ndingasetofnodeswithminimumsizewhichcausesthedesiredimpact.Dependingonthecascadingmechanism,wedesigndifferentstrategiestosolvetheproblemefcientlybyexploitingboththepropertiesofboththecascadingmechanismandthenetworkstructure. 10

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CHAPTER1INTRODUCTIONNowadays,theworldismoreandmoreconnectedandwecanseenetworkseverywhere.Networks,frompowersubstationsinpowernetworks,routersincommunicationnetworks,usersinonlinesocialnetworks,etc.,representtheinteractionbetweenentitiesandplaycrucialrolesintheeconomy.Powernetworksareimportantinfrastructurenetworkswhosemalfunctioncanleadthechangeorstoppingofalmostdailyactivities.Ontheotherhand,large-scaledonlinesocialnetworksareeasingthecommunicationbetweenpeoplebyprovidingaplatformforuserstoconnectandkeepupdatingfromeachother.Duetothehighimpactofthesenetworksontheeconomy,itiscrucialtostudyphenomenawhichsignicantlyaffectactivitiesinnetworks.Asentitiesinnetworksinteractwitheachother,thecascadingpropagationisoneofthemostnoticeablephenomenainnetworks.Ifaneventhappensataparticularentity,itcantriggerseventsatotherentitiesthroughtheconnectionbetweenentitiesinthenetwork.Forinstance,theinteractionbetweenusersinonlinesocialnetworksservesasthemediumtospreadinformation,ideas,andinuences.Initially,onlyasmallgroupofusersawareoftheinformationandshareinthenetworks;then,theinformationisspreadtotheirfriends,friendsoftheirfriends,andsoon.Asaresult,largenumberofuserswillawareoftheinformationevenbeforethemassmediabroadcastitasinthecaseofMichaelJackson'sdeath[ 1 ].Inthepowernetwork,thecascadingpropagationcancauseseveredamagebymultiplyingtheinitialfailure.In2003,theinitialfailureofonepowerlinetriggeredaseriesoffailureswhichresultedintheoutageofthemajorityofItaly[ 47 ].Thelarge-scaledeffectofthecascadingpropagationinspiresustodesignmethodstoexploititspositiveeffectsandprohibitnegativeeffects.However,thecascadingmechanismsarevariousinnetworksandtoobroad,thuswefocusoninvestigatingthecascadingpropagationinfollowingsettings: 11

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1.1CascadingFailureinaNetworkNetworkswheretheoperationofanodestronglydependsontheoperationofothernodeslikepowernetworksareextremelyvulnerableundercascadingfailure.Inthesenetworks,everynodebearsdynamicoperationalloaddependingonthedemand.Ifthedemandishigh,thenodes'loadsarehighandmayreachtotheirmaximumoperationalcapacities.Ingeneral,nodessharethenetworkdemandsothateachofthemcanoperateunderthepermittedcapacity.However,whensomenodesaremalfunctionedorfailed,theyshifttheloadtonearbynodesinthenetwork.Thesenodesmaybeforcedtoworkbeyondtheircapacitiessotheyareoverloadedredistributetheirloadontoothernodes.Asaconsequence,alargenumberofnodesmaybeoverloadedandtherebythenetworkhaltstheoperationentirely.Asthefailureofasmallgroupofnodesmayresultinacatastrophicdamageonthenetworkoperation,itisimportanttoidentifysuchgroups.Thesenodesarecriticaltotheoperationofthenetwork,thusweneedtoprotectthemfrombeingattacked.Althoughexistingliteratureprovidesvariousvulnerabilityassessmentofnetworksunderthecascadingfailure,thereisstilllackingofefcientsolutionsforthisproblem.Theseworksmainlyexploitthecentralitymeasurementtolocatemostcriticalnodeswhicharenotenoughtocapturethecomplicatedinteractionofnodes.Wedesignnewmethodswhichconsiderbothnetworkstructureandtheinteractionbetweennodestoprovidebettersolutions. 1.2CascadingFailureinInterdependentNetworksInreality,infrastructurenetworksareinterdependentoneachotheratalargedegree.Thepowerstationsinthepowergridconsumethefueldeliveredbythetransportationnetworktogenerateelectricityandarecontrolledviathecommunicationnetwork.Ifthetransportationnetworkorthecommunicationnetworkencountersanyproblems,theoperationofthepowernetworkwillsuffered.Ontheotherhand,thepowernetworkprovideselectricityforroutersofthecommunicationnetworkandelectrical 12

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vehicles.Therefore,thefailureofacriticalgroupofnodesinanynetworkmayresultinaseriesofcascadingfailuresacrossthesystemandcausecatastrophicloss.Ifeachnetworkistreatedseparately,wewillunderestimatethevulnerabilityofnetworks.Weneedtorevisitthevulnerabilityassessmentofnetworkstakingintoaccounttheeffectofinterdependenciesbetweennetworks.Let'sconsiderthepowerandcommunicationnetworks.Intheattackingpointofview,anattackercananalyzetheinterdependenciesandidentifynodeswhosefailurestriggeralarge-sizedcascadingprocessbackandforthbetweennetworks.Ifweonlyinvestigateasinglenetwork,thesenodesseemtobescot-freeandwefailtoprotectthem.Althoughtherearemanyefcientmethodstoidentifycriticalnodesinasinglenetwork,itisstilllackingonesforinterdependentnetworks.Inthiswork,weproposeanewcentralityforinterdependentnetworkswhichcanbeusedtolocatecriticalnodesefciently. 1.3InuenceDiffusioninMultipleOnlineSocialNetworksIntheareaofonlinesocialnetworks(OSNs),thecascadingpropagationofinformationtransformsnetworkssuchasFacebook,Google+,andTwittertoafruitfulfoundationforviralmarketing.Thesenetworksequipuserstoolstoconnectandmakenewfriends,toshareopinions,toupdateinformationfromfriends,etc.,thusattractaconsiderablefractionthepopulationtojoinin.Inreturn,userscreatethecontentandcirculatetheinformationatalevelthathasbeenachievedbeforebyanyofpreviouscommunicationmedium.Inaddition,usersinonlinesocialnetworksalsoincurthesamepeer-pressureeffectasthereality,i.e.,inwhichanindividual'sopinionordecisionisinuencedbyhisfriendsandcolleagues.ThesefactorsraiseapracticalimportantprobleminOSNs:howtondthesmallestsetofinuencerswhocaninuenceamassivenumberofusers.AnoticeablepropertyOSNsistheoverlappingamongmajorOSNswhichhasastrongimpactonthediffusionofinformation.Sinceausercansharetheinformationinallnetworkswhichheparticipatesin,theinuenceofauserinallnetworksissignicant 13

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largerthaninanynetwork.Therefore,itisessentialtoevaluatetheinuenceofusersinmultipleOSNstoidentifymostinuentialones.However,wecannottriviallymitigateevaluationmethodsforasinglenetworktomultiplenetworks.Toovercomethisdifculty,weproposenovelschemestocouplemultiplenetworksintoonenetworkreservingalldiffusioninformationandsolvetheprobleminthecouplednetwork. 1.4OrganizationTherestoftheworkisorganizedasfollows.Chapter 2 studiesthevulnerabilityofpowernetworksundertheloadredistributionmodel.Inchapter 3 ,wepresentthecascadingfailuremodelforinterdepentnetworksandproposealgorithmstodetectcriticalnodes.Next,chapter 4 investigatesinformationdiffusioninmultipleonlinesocialnetworks.Finally,chapter 5 concludesthewholethesis. 14

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CHAPTER2CASCADINGFAILUREUNDERLOADREDISTRIBUTIONINNETWORKSTheimportantroleofpowernetworksintheeconomyaswellasinthesocietyhasattractedagreatdealofresearchefforttoanalyzethevulnerabilityofthesenetworks.Thefailureormalfunctionofthesenetworkscancausesevereeffect.On28September2003[ 47 ],thewideareablackoutaffectedthemajorofItalyandmade3/4ofItalywithoutelectricityfor2hours,thetrafcsystemishalted.Thisshowsthatlargeblackoutsarenotraredandcanhappeneverywhere.Moreover,itimpliesthatintentionalattackscancausemassdamagetopowernetworks.Whenthesmallnumberofcomponentsareattacked,thelargeblackoutcanhappeninaveryshorttime.Thus,itiscrucialtoidentifymostvulnerablecomponentsofthepowernetworksothatwecanprotectinadvance.Thecommondenominatoroflargeblackoutsisthatthefailuresofcomponentshappenedaccordingtothecascadingmanner.Itoftenstartswiththefailureofoneorafewcomponents,thensomeothercomponentsarefailedduetothedependencieswithpreviousfailedcomponents.Thefailureofthesecomponentscontinuetocauseothercomponentsfail.Theprocesscontinuesuntilthereisnomorefailedcomponent.Thepowerstatiosnwhicharenodesinthepowernetworkcanonlyworkwelliftheloadisunderthemaximumcapacitytheycanhandle.Whenastationisoverloaded,itcannotworkwiththebestperformanceorevenfails.Duringtheoperation,thepowernetworkisdesignedsuchthatallstationsworkundertheircapacity.Butwhensomestationsfails,otherstationswhicharedirectlyorindirectlyconnectedwithfailedonesmayhavebearmoreload.Iftheloadofastationsurpassesitscapacity,itwillfailandcontinuetoshreditsloadtootherstations.Asaresultoftheloadredistributionprocess,alargenumberoffailedstationsmaybefailedattheend.Wewouldliketopredicttheprocesssothatwecanpreventit,butthedependenciesbetweenstationsmakeitdifculttodoso.Thus,it 15

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isnecessarytodevelopefcienttoolstoanalyzethesophisticatedcascadingprocessoffailuresinpowernetworks.Inthischapter,westudythecriticalnodedetectionprobleminpowernetworksundertheloadredistributionofnodes.Specically,weaimtondthesetofknodeswhosefailuresmaximizethenumberfailednodesafterthecascadingfailure.Wedesigntwoefcientalgorithmstosolveproblem:adaptivecascadingpotentialalgorithmandcooperatingattackalgorithm.Theefciencyofeachalgorithmdependsonthetopologicalstructureofthenetwork,hencetheycompensateeachothertosolvetheproblem.Therestofthechapterisorganizedasfollows.WerstpresenttheloadredistributionmodelandproblemformulationinSection 2.1 .Section 2.2 showsthehardnessresult.Afterthat,weproposethecascadingpotentialmetricanddesignvariousalgorithmsinSection 2.3 .WenextintroducethecooperatingalgorithmwhichisefcientonrobustnetworksinSection 2.4 .Section 2.5 showstheexperimentalevaluation.Finally,wereviewtheliteratureinSection 2.6 andsummerizethechapterinSection 2.7 2.1NetworkModelandProblemFormulation 2.1.1GraphNotationsThenetworkismodeledbyaweighteddirectedgraphG=(V,E)withvertexsetVofjVj=nverticesandedgesetEofjEj=morientedconnectionsbetweenvertices.Eachedge(u,v)isassociatedwithaweightw(u,v)presentingtheoperatingparameterofthenetwork.Thehigherw(u,v)is,themoreloadisdistributedfromutov.Inaddition,eachvertexuhasthecurrentloadL(u)andacapacityC(u).ThecapacityC(u)isthemaximumloadthatvertexucanaccept.Wedenotethesetofincomingneighbors,outgoingneighborsofubyN)]TJ /F8 7.97 Tf -0.93 -7.29 Td[(uandN+u,respectively. 2.1.2CascadingFailureModelIntheLoadRedistributionmodel(LR-model)[ 56 ][ 53 ],nodesarefailedinthecascadingmannerduetotheloadredistributionoffailednodes.Initially,asetofnodes 16

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Sarefailed,thenthefailuresarepropagatedtoothernodesintimesteps.Whennodeufails,itsloadisredistributedtoitsneighborsasillustratedinFig. 2-1 .Eachaliveneighborwillreceivedanadditionalloadwhichisproportionaltoitsweight.Precisely,eachneighborvofuwillreceiveadditionalload:L(v)=L(u)w(u,v) Pz2N+uw(u,z)Duetotheloadredistribution,theloadofsomenodesareexceedingtheircapacities,hencefailinthenexttimestep.Theprocessofloadredistributionandnodefailingwillstopwhentherearenomorefailednodes.ThesetoffailednodescausedbytheinitialfailureofSisdenotedbyF(S). Figure2-1. Whennodeufails,itsloadisredistributedtotheneighbornodes.Amongthesenode,vreceivesahighportionofloadfromuandbecomesoverloaded.Theloadofvisredistributedtoitsneighborswhichmakeszfail.Finally,theloadfromzcontinuestocausewfailandtheprocessstops. 17

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2.1.3ProblemDenitionDuetothecascadingfailures,thefailuresofasmallsetofnodesScanresultinacatastrophicnumberoffailednodes.Thesenodesbecomesthetargettoattackthenetwork.Additionally,giventhesamesetofattackednodes,differentattackingordersleadtodifferentoutcomes.Withthesameattackingcost,theattackercanchoosethebestorderwithsuitabletimeforeachattackednode.However,thecascadingfailureshappenveryfast,itisalmostimpossibletoschedulethefailureofeachnodewithspecictimesteps.Weconsideramorepracticalstrategyinwhichtargetnodesareattackedonebyone.Thenextnodeistakendownwhenthecascadingprocessstops.Inparticular,givenanordersetS=fs1,s2,...,skg,thesetoffailsaftersiisattackedisFi(S)=F(Fi)]TJ /F9 7.97 Tf 6.59 0 Td[(1(S)[fsig).DenoteF+(S)asFk(S),thesetoffailednodeswhennodesinSisattackedserially.Weformallydenetheproblemasfollows. Denition1(CascadingCriticalNodeProblem(Cas-CNP)). GivenanetworkG=(V,E)andanintegerk,theproblemaskstondaorderedsubsetSVofsizejSj=ksuchthattheserialfailuresofnodesinSmaximizesthenumberoffailednodesF+(S)undertheLR-model. 2.2InapproximabilityResultInthissection,weshowthealgorithmichardnessoftheCas-CNproblem.Weexpecttodesignanalgorithmthatcanidentifytheoptimalseedsetinanacceptabletime.However,itmaytakethetimeasanexponentialfuntionofthenumberofnodestocomputeevenasetwhoseimpactisclosetotheoptimalset's.ThehardnessresultisshowninTheorem 2.1 Theorem2.1. ItisNP-hardtoapproximatetheCasCNproblemwithinratioofO(n1)]TJ /F11 7.97 Tf 6.59 0 Td[()foranyconstant1>>0. Proof. Weusethegap-introductionreduction[ 51 ]toprovetheinapproximabilityoftheCasCNproblem.UsingapolynomialtimereductionfromSetCover,totheCasCNproblem,weshowthatifthereexistsapolynomialtimealgorithmthatapproximatesthe 18

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laterproblemwithinO(n1)]TJ /F11 7.97 Tf 6.59 0 Td[(),thenthereexistsapolynomialtimealgorithmtosolvetheformerproblem. Denition2(SetCoverproblem). GivenauniverseU=fe1,e2,...,eng,acollectionofsubsetsS=fS1,S2,...,Smg2U,andanintegerk,theSetCoverproblemaskswhetherornotthereareksubsetswhoseunionisU.InsteadofusingthehardnessresultofthegeneralSetCoverproblem,weusetheresultonarestrictedvariantMIN3SC2oftheSetCoverproblemwherethesizesofsubsetsareatmost3andeachelementappearsinexactlytwosubsets. Theorem2.2([ 20 ]). TheSetCoverproblemisNP-hardevenwhenthesizesofsubsetsareboundedby3andeachelementappearsinexactlytwosubsets.Reduction.GivenaninstanceoftheSetCoverproblemI=(U,S,k)whereeachelementappearsinexactlytwosubsets,n1=jUjandm1=jSj,weconstructaninstanceI0oftheCasCNproblemasillustratedinFig. 2-2 Figure2-2. ReductionfromMIN3SC2toCasCN 19

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ThevertexsetV.AddasetvertexuiforeachsetSi2S,anelementvertexvjforeachelementej2U,andd=(n1+m1)2 extraverticesq1,q2,...,qd.TheedgesetE.Addedge(ui,vj)iftheelementvjinthesetSi.Inaddition,thereisanedgefromvjtoqp81jn1,1pd.Alledgeshavetheweightof1.Vertexloadandcapacity.TheloadandcapacityofthesetvertexuiareL(ui)=jSijandC(ui)=1+L(ui).TheloadandcapacityoftheelementvertexvjareL(vj)=d)]TJ /F5 11.955 Tf 12.13 0 Td[(1andC(vj)=d)]TJ /F5 11.955 Tf 11.95 0 Td[(0.5.Allextraverticeshavetheloadof0andcapacityofn1)]TJ /F5 11.955 Tf 11.96 0 Td[(1+n1 d.Next,weprovethatifIhasasetcoverofsizekthenthereexistaseedingsetAVsuchthatjF(A)j>d.Otherwise,forallAVandjAjk,jF(A)jC(qp),henceallextraverticesarefailed.ThecascadingprocessstopswithjF(A)j=d+n1+kfailedvertices.InthecaseIhasnosetcoverofsizek,wewillshowtheoptimalseedingsetcancauseatmostn1+knodesfailinI0.LetAbeanarbitraryoptimalseedingset.Weobservethatasetvertexonlyfailswhenitisselectedintheseedingsetsinceithasnoincomingedge.Thus,thereareatleastm1)]TJ /F3 11.955 Tf 12.4 0 Td[(ksetverticeswhicharenotinA.Wecanreplaceanyextravertexqp2A,ifthereareany,byaunselectedsetvertexwithoutdecreasingthenumberoffailednodes.Next,supposethatthereexistsanelementvertexvj2S,wecanalsoreplaceitbyasetvertex.Ifvjisadjacenttosomevertexui2A,wecanremovevjfromAwhilemaintainingthesamenumberoffailednodes.IfvjisnotadjacenttoanyvertexinA,wejustreplacevjbyoneofitsneighborhood 20

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setvertexui.uiwillmakevjfail,sothenumberoffailednodescausedbyAisnotdecreased.So,wecanreplaceextraandelementverticesinAsuchthatAcontainsonlysetvertices.Since,thereisnosetcoverofsizek,thereatleastonevertexisnotadjacenttoanyvertexinA,i.e,thenumberoffailedelementverticesisatmostn1)]TJ /F5 11.955 Tf 12.3 0 Td[(1.Eachfailedelementvertexvjisadjacenttoatmost2setvertices,henceitsloadisatmostL(vj)d+1.Eachextravertexqpreceivesatmost(d+1)=dredistributedloadfromfailedelementverticeswhichareaccumulatedtoatmost:(n1)]TJ /F5 11.955 Tf 11.96 0 Td[(1)(d+1) d=n1)]TJ /F5 11.955 Tf 11.95 0 Td[(1+n1)]TJ /F5 11.955 Tf 11.95 0 Td[(1 dd (d+m+n)1)]TJ /F13 5.978 Tf 5.75 0 Td[(>d (2d)1)]TJ /F13 5.978 Tf 5.76 0 Td[(>d 2=(m1+n2)2 2>m1+n1Ontheotherhand,ifIhasnosetcoverofsizek,thentheoptimalseedingsetAoptofI0causeslessthan(m1+n1)verticesfail.Wehave:jF(A(I0)jjF(Aopt)j<(m1+n1) 21

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ItimpliestheIhasasetcoverofsizekifandonlyifjF(A(I0)j>(m1+n1).Hence,wecanuseAtodecidetheSetCoverprobleminpolynomialtimei.e.P=NP. 2.3CascadingPotentialandDerivedAlgorithmsInthissection,weintroduceanewmetrictomeasurethenodeimportanceunderthecascadingfailure,andthenapplyittodesignefcientalgorithmsforCasCN.Toevaluatethevulnerabilityofnetworksundertheloadredistribution,previousworksintheliteratureproposevariousrankingmethodsandmeasuretheeffectofattackingtopknodes.However,thesemethodsconsiderverylimitedtopologicalinformation,hencemaymissthemostcriticalnodes.In[ 7 36 53 ],theauthorssolelyusetheloadasthecriteriontoranknodes.Thefailureofahighloadnodeintuitivelytendstocausealargenumberofnodesfailasitredistributesalargeamountoflargetoitsneighbors,butcascadingfailuresstartedfromasmallloadnodeattherightpositionmayresultinalargernumberoffailednodes[ 54 ].Wangetal.[ 54 ]overcomethisshortcomingbydirectlyassessingtheeffectofthecascadingprocess,thenumberoffailednodes,whichistriggeredbytheevaluatednode.Nevertheless,thedirectimpactisonethetopfactors,theyfailtoincorporatetheindirectimpactintothenodeimportance.Whenmultiplenodesareattackedinthenetwork,theindirectimpactofanodeisthebaseforthedirectimpactofothernodes.Next,weintroduceanewmetricwhichconsidersbothdirectandindirectimpactofthenode. 2.3.1CascadingPotentialThecascadingpotentialofanodeisdenedascombinationofallpossibleimpactsanodecausesinthenetworkunderthecascadingeffect.Let'sconsiderthefailureofnodeu.Foranyothernodev,therearetwopossibleimpactsthatucaninduceonv: Failureimpact.Thefailureofuleadstothefailureofv. Loadimpact.Thefailureofumakestheloadofvincreasebutnotenoughtofail. 22

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Theoverallfailureimpactandloadimpactofuinthenetworkaredenedasthenumberoffailednodesandthetotalofincreasedloadofunfailednodes,respectively.Thecascadingpotentialofuisthelinearcombinationofthesefactors:C(u)=jF(fug)j n+Pv2V)]TJ /F8 7.97 Tf 6.58 0 Td[(F(fug)Lu(v) Pv2V)]TJ /F8 7.97 Tf 6.59 0 Td[(F(fug)(C(v))]TJ /F3 11.955 Tf 11.95 0 Td[(L(v))whereF(fug)isthesetoffailednodeswhenufailsandLu(v)istheadditionalloadthatvreceivesduetothefailureofu.Inthisformula,wenormalizeboththefailureandloadimpactstoavoidtheunitdifference.Thefailureimpactisdividedbythenumberofnodes,henceisatmost1whenallothernodesfail.Similarly,theloadimpactisdividedbythetotalofcapacity-loaddifferenceofunfailednodesandachievesthemaximumvalue1whenallremainednodesareattheedgeoffailure,i.e.,themostvulnerablestateofthenetwork.Theroleoftheloadimpact.Intheformulationofthecascadingpotential,theloadimpactplaysanimportantroletoprovideabetterassessmentofthenetworkvulnerabilitycomparingtothemetricin[ 54 ].Ifonlyonenodeisattacked,itisobviouslytochoosethenodewhichmaximizesthenumberoffailednodes,i.e.,touseWangetal.'smetric.However,whenmultiplenodesareattacked,weneedtoconsidertheco-impactofattackednodestotriggeralargesizecascadingfailure.Theloadimpactisbridgeconnectingtheimpactofthesenodessincetheloadimpactofanodeisthebaseforthefailureimpactofothernodes.Forexample,ifuhasthemaximumloadimpactof1andthenetworkisstronglyconnected,thenattackinganynodeafterucantakedownthewholenetwork.Thus,thecascadingpotentialevaluatetheimportanceofnodesmorecomprehensively. 2.3.2CascadingPotentialAlgorithmIntuitively,wecanusecascadingpotentialdirectlytodesignanalgorithmforCasCN.Werstcomputethecascadingpotentialofallnodes,thenselecttopkasattackednodes.ThealgorithmisdescribedinAlgorithm 1 23

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Algorithm1CascadingPotentialAlgorithm Require: AnetworkG=(V,E),anintegerk. Ensure: AsetSofkattackednodes. Computethecascadingpotentialofallnodes Sortnodesinnon-increasingorderofthecascadingpotentialC(u1)C(u2)...C(un) InitializeS ; j 1 fori=1tokdo S S[fuig endfor ReturnS Timecomplexity.IttakesatmostO(m)tocomputethecascadingpotentialofeachnode.Thus,thetotalrunningtimeisO(nm+nlogn). 2.3.3AdaptiveCascadingPotentialAlgorithmTheCascadingPotentialalgorithmrunsfast,butitneglectsanimportantpropertyofthecascadingfailure:theoverlappedimpactofselectednodes.Letconsidertwonodesuandvwhichbothhavefailureimpactonnodez.Ifuisselectedbeforev,thenvhasnoimpactzaszisalreadyfailed.Asaconsequence,somenodeshavehighimpactinitiallywillhavesmallimpactatthelateoftheselectionprocess.Wecanimprovetheperformanceofthealgorithmbyupdatingtheimpactofremainednodesonthey.Morespecically,attheithiteration,theimpact(failureorloadimpact)ofnodeuonfailednodes(duetotheselectionofrsti)]TJ /F5 11.955 Tf 12.67 0 Td[(1attackednodes)willbesubtractedfromtheinitialimpactofu.Afterthat,thenodewithhighestremainedimpactwillbeselected.Thecrucialproblemishowtoupdatetheimpactofnodesefciently.Wemaynaivelykeepthelistofimpactednodesforeachnodeu.Ateachiteration,wecomparethelistofimpactednodesandthesetoffailednodestoupdatethesubtracttheimpactonfailednodes.Thiscanresultin(n3)runningtimeforeachiterationwhichisverytimeconsuming.Wereducetheupdatingtimebyreversingtheprocess.Eachnodevwillkeeptwolistsofnodes:thelistFI[v]containsnodeswhichhavefailureimpactonvandthelistLI(v)containsnodeswhichhaveloadimpactonv.Sincetheloadimpact 24

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Algorithm2AdaptiveCascadingPotentialAlgorithm Require: AnetworkG=(V,E),anintegerk Ensure: AsetSofkattackednodes. foreachv2Vdo InitializeFI[v] ;,LI[v] ; endfor foreachu2Vdo ComputeC(u) foreachv2F(fug)do FI[v] FI[v][fug endfor foreachv:Lu(v)>0andv=2F(fug)do LI[v][u] Lu(v) Pz2V)]TJ /F8 5.978 Tf 5.75 0 Td[(F(fug)(C(z))]TJ /F8 7.97 Tf 6.59 0 Td[(L(z)) endfor endfor InitializeS ; foreachu2Vdo Mark[u] False endfor fori=1tokdo u argmaxv2VnF+(S)fC(v)g S S[fug foreachv2F(S)do ifMark[v]==Falsethen Mark[v] True foreachu2FI[v]do C(u) C(u))]TJ /F5 11.955 Tf 11.95 0 Td[(1=jVj endfor foreachu2LI[v]do C(u) C(u))]TJ /F3 11.955 Tf 11.95 0 Td[(CL[v][u] endfor endif endfor endfor ReturnS ofothernodesonvaredifferent,weuseLI[v][u]tostoretheloadimpactofuonvafterthenormalization.Whenvisfailed,theimpactofnodesinitslistswillbeupdated.Thecrucialpointisthateachnodeonlyfailsonce,thustherunningtimeisreducedsignicantly.ThealgorithmwithadaptivecascadingpotentialisdescribedinAlgorithm 2 25

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Timecomplexity.Sinceeachnodehasimpactonatmostnnodes,thetotalsizeofallFIandLIlistsareatmostn2.ThenumberofupdatesisboundedbythetotalsizeofFIandLIlists.ThereforethetotalrunningtimeisO(nm+n2+kn). 2.3.4FullyAdaptiveCascadingPotentialAlgorithmOnthelineofcascadingpotentialbasedalgorithms,wecontinuetoimprovethesolution'squalitybyspendingmoretimetocalibratethecascadingpotentialofnodes.Afteranodeuisattacked,thenetworkstateischangedwithnewfailednodesandloadupdates;andthismaydecrease(asdiscussedintheproceedingpart)orincreasetheimpactofanode.Thefailureofuaddsloadtomanynodesandmakesthemmorevulnerable.Althoughtheimpactofaremainednodevisdeductedbytheimpactonfailednodes,itcanstillincreasesinceothernodesareeasiertobefailed.Wecanfullyupdatethecascadingpotentialofeachnodeasfollows.Afterselectinganewnode,wesimulatethecascadingfailuretriggeredbyitandobtainanewgraphofremainednodes.Inthisgraph,theloadofanodeistheloadwhenthecascadingprocessstops.Wethencanevaluatethecascadingpotentialofallnodesintheupdatedgraphandselectonewithhighestvalue.WepresentthealgorithminAlgorithm 3 Algorithm3FullyAdaptiveCentrality Require: AnetworkG=(V,E)andanintegerk. Ensure: AsetSofkattackednodes. InitializeS ; fori=1tokdo ComputethecascadingpotentialofallnodesinG Selectuasthenodewithhighestcascadingpotential S S[fug UpdatenodeloadsandremoveallfailednodesinGwiththefailureofu endfor ReturnS Timecomplexity.WeneedtocomputethecascadingpotentialofallnodestoselectanewonewithtimeO(nm).ThusthetotalrunningtimeisO(kmn).However,the 26

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algorithmmayrunmuchfasterthantheworstcasetimesincethesizeoftheupdatedgraphdecreaseswhenanewnodeisselected. 2.4CooperatingAttackAlgorithmThekeytoconnecttheimpactofmultiplenodesintheabovealgorithmsistheloadimpactwhichisaconnectionlinkwhenthenetworkisrobust.Inthiscasewherenodesarehighfailuretolerant,i.e.,thegapbetweenthecapacityandloadisbig,thefailureimpactofeachnodeissmall.Thus,nodeswithhighloadimpactstendtobeselected.concentrateIftheloadofthesenodesarescatteredtomanynodes,theyarenotlinkedtogethertomakeothernodesfail.Asaconsequence,therearealargenumberofnodeswhoseloadsareincreased,butthereisonlyafewfailednodes.Weincidentallytrytomaximizethetotalloadimpactinsteadofthefailureimpacttheobjectivefunction.Weneedabetterstrategywhichbuildsastrongconnectionbetweenselectednodestoincreasethenumberoffailednodes.Tofulllthisgoal,thenewstrategyshouldsatisfyfollowingfeatures: Theredistributedloadofselectednodesshouldbeconcentratedoncertainnodestofailthem.Ifearlyselectednodesredistributedloadtoasetofnodes,thenlaterselectednodesshouldalsoredistributeloadtothisset.Itissaidthatselectednodesarecooperatinginredistributingloadtomakemorenodesfail. Selectednodesshouldcooperatetomakehighloadnodesfail.Thefailureofhighloadnodescanexpandthecascadingfailurefurther.However,ifhighloadnodepreferencereducesthenumberoffailednodes,thenewstrategyshouldavoidblindlyfavoringtofailhighloadnodes.Wedesignanewevaluationfunction,theefciency,ofnodeswithpropertiesthattailortheselectionprocesstoembracebothdesiredfeatures.Firstly,wegivehigherevaluationtonodeswhichredistributesitsloadtoload-increasednodes.IfthefailureofupushesanadditionalloadLu(v)onv,thentheimpactofuonvisdenedby:(u,v)=Lu(v) C(v))]TJ /F3 11.955 Tf 11.96 0 Td[(L(v) 27

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whenLu(v)+L(v)C(v).SinceitrequiresC(v))]TJ /F3 11.955 Tf 11.97 0 Td[(L(v)additionalloadtomakevfail,wecaninterpretthatumakesafractionLu(v) C(v))]TJ /F8 7.97 Tf 6.58 0 Td[(L(v)ofvfail.Thenewimpactfunctionimpliesthatifthemoreloadanodereceives,themorelikelythenewselectednodewillredistributeloadonit.Ontheotherhand,theevaluationofuishigheriftheloadsofitsneighborsareincreased.ThisimplicationisstatedinProposition 2.1 Proposition2.1. Foranynodevattwopointsoftime,ifvreceivesmoreloadatthesecondtimepoint,i.e.,L2(v)>L1(v),thentheimpactofothernodeuwiththesameredistributedloadLishigheratthesecondtimepoint:2(u,v)1(u,v). Proof. Wehave:2(u,v)=L C(u))]TJ /F3 11.955 Tf 11.96 0 Td[(L2(u)>L C(u))]TJ /F3 11.955 Tf 11.95 0 Td[(L1(u)=1(u,v) NotethatweassumeuredistributethesameloadonvintheProposition 2.1 ,i.e.theloadofuisthesameattwopointsoftime.Infact,theloadofumayincreaseduetotheselectionofpreviousnodes,thustheevaluationofuevenincreasesmoreatthesecondpointoftime.Tofulllthesecondfeature,weassignhighervaluestohighloadnodeswhichareimpacted.ThevalueofanodewithloadLis:(L)=eL 1+eLThefunction(L)ismonotoneincreasingandintherange0.5(L)<1.Themonotoneincreasingofthefunctionshowsthepreferencetowardhighloadnodes.Recallthatthemaingoalistoincreasethenumberoffailednodes,soevennodeswiththelowestloadhavethevalueatleasthalfofthehighestvaluenodes.Next,wewilldenetheefciencyofselectinguviatheimpactonv.Intuitively,umakes(u,v)fractionofvfailandvhasvalueof(L(v)),thustheefciencyofu 28

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representedonvis:(u,v)=(u,v)(L(v))Finally,weobviousshouldtakeintoaccountthenumberoffailednodeswhenevaluatingnodeu.Theoverallefciencyofuisthetotalofthenumberoffailedandtheefciencyonunfailednodes:(u)=jF(fug)j+Xv2VnF(fug)(u,v)Theefciencyevaluationshowsseveralnotablepropertieswhichserversourdesigngoalasfollowings:Increasethenumberoffailednodesrst.Ifumakeszfailandhasefciency(u,v)ontheunfailednodev,thenthecontributionofztotheoverallefciencyofuisalwayshigherthanvsince1(u,v)(L(v)).Avoidredistributingloadtoimpossible-to-failnodes.Ifnodevneedstoomuchadditionalloadbeforefailing,itwillbeignoredinefciencyevaluationofnodesasstatedintheProposition 2.2 Proposition2.2. GiventwonodesuandvwithxedloadL(v),theefciencyofuonvismonotonedecreasingandgoesto0whenthecapacityC(v)ofvincreasesandgoestoinnity. Proof. Itiseasytoseethat(u,v)ismonotonedecreasingandgoesto0whenC(v)increasesandgoestoinnity.Inaddition,(L(u))isaconstant,sotheefciency(u,v)=(u,v)(L(v))decreasesandgoesto0. Notfavoringhighloadnodeswithallcost.Weconsiderthecasethecapacityislineartotheload,acommonsettingintherealitytoguaranteethesafetyofnodes.Inthiscase,eventheloadofnodevisextremelylarge,itisstillignoredasshowninPreposition 2.3 29

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Proposition2.3. SupposethatthecapacityC(v)islineartotheloadC(v)=TL(v)withconstantfactorT.Then,theefciencyofanynodeuonvgoesto0whentheloadL(v)goestoinnity. Proof. Wehave:(u,v)(L(u))=Lu(v) C(v))]TJ /F8 7.97 Tf 6.59 0 Td[(L(v)eL(v) 1+eL(v)
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strategiessortnodesbasedonsomecriterionandselecttopknodesasattackednodes.Thesortingcriteriaare: Largestload(HL). Lowestload(LL). Highestpercentageoffailure(PoF).Thepercentageoffailureofanodeuisthefractionofnodesisfailedwhenufails. Highestriskiffailure(RIF).RIFofanodeuistherationbetweenitsloadandthetotalloadofitsneighbornodes.WeomittherequiredredundancyofWangetal.[ 54 ]sinceitprovidesthesameorderofnodesasRIF.Afterthat,weevaluatetherobustnessofnetworksunderdifferentfailuretoleranceschemestoidentifythesuitablenetworkdesignconsideringtheeffectofcascadingfailures. 2.5.1DatasetsRealnetwork.WeusetheWesternNorthAmerican(WNA)powergridnetwork[ 55 ]with4941substationsand6594transmissionlinestorunexperiments.However,thedatasetislackingofloadandcapacityinformationofnodes,thusweusethemethodin[ 56 ]toassigntheloadandcapacityforeachnode.TheinitialloadofofnodeuisgivenbyL(u)=d(u)whered(u)isthedegreeofuandisatunableparameter.Thisassignmentmethodisreasonableastheloadofthenodeisshowntobescaledwithitsdegree[ 59 ].Vertexcapacitiesareassignedbasedonthreedifferentschemes.Normalnetworks.Innormalnetworks,thecapacityC(u)ofeachnodeuisproportionaltoitsinitialloadL(u):C(u)=TL(u)whereTisaconstantrepresentingthesystemtolerance.ThehigherTis,themoreendurancethenetworkisunderthecascadingfailure. 31

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Safenetworks.Insafenetworks,thenodecapacitiesareassignedintwophases.First,thecapacityC(u)ofeachnodeuisscaledasthenormalnetwork,i.e:C(u)=TL(u)Then,thencapacitiesofallnodesareraisedtosatisfytheN)]TJ /F5 11.955 Tf 9.66 0 Td[(1failuretolerancecriterioninwhichthefailureofanynodewillcausenoadditionalfailednodes.Itmeansthatanynodeuwillnotfailwhenitreceivestheredistributedloadfromanyofitsneighbor.Thecapacityofuwillbe:C(u)=maxfC(u),maxv2N)]TJ /F9 7.97 Tf 6.26 -2.27 Td[((u)fL(u)+w(v,u)L(v)ggScaledSafenetworks.Incontrasttosafenetworks,scaledsafenetworksareformedbyraisingthenodecapacitiestosatisfyN)]TJ /F5 11.955 Tf 11.81 0 Td[(1failuretolerancecriterionrst,thenbescaleduplater.Inparticular,thenetworkismadesafebyassigningthecapacityofeachnodeuas:C(u)=maxv2N)]TJ /F9 7.97 Tf 6.26 -2.27 Td[((u)fL(u)+w(v,u)L(v)gThenscaleupthecapacityofutoC(u)=TC(u).SynthesizedNetworks.WealsoruntheexperimentsonsynthesizednetworksgeneratedbyErdos-Renyirandomnetworkmodel[ 26 ].Eachnetworkhas5000verticeswiththeaveragedegreeof4or8.Theotherparametersofthenetworkisgeneratedsimilartoaboveschemes. 2.5.2TheperformanceofDifferentAlgorithmsWerstcomparetheperformanceofproposedalgorithmsandthepreviousworks.Werunexperimentsonnetworkswithdifferentsystemtolerancevaluesand=1.TheresultsareshowninFig. 2-3 2-4 ,and 2-5 .Whennetworkshavesmalltolerancevalues,theyaresovulnerableunderanyattackingstrategy.Thefailureofoneortwonodescanleadtothefailureofalmostthewholenetwork.Ontheotherhand,whenTishigh,thenetworkcanenduremultipleattackswithoutfailing.Theseguresalsoshow 32

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AWSNnetwork BSynthesizednetwork(d=4) Figure2-3. VulnerabilityofWSNnetworkundernormalsetting thatpreviousattackingstrategiesdonotworkwellonsafenetworks.Onsafenetworks,theperformanceofFACPandCAalgorithmsarethebestsincetheycanadapttothechangeofnetworkstatustochoosenodesforattacking.Additionally,whenthenetworkisvulnerable,FACPshowsbetterperformance.Earlyattackednodespushremainingnodestotheboundaryoffailure.Theredistributedloadoflaterattackednodescanmakethemfaileasily.Thesituationischangedwhennetworksarerobust.CAalgorithm 33

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optimizesthenumberoffailednodesateachiteration,soitsproducedseedsetmakemorenodesfail. AWSNnetwork BSynthesizednetwork(d=4) Figure2-4. VulnerabilityofWSNnetworkundersafetysetting 2.5.3NetworkRobustnessUnderDifferentSettingsWeobservedthatscaledsafenetworksaremorerobustnessthansafenetworksandsafenetworksarestrongerthannormalnetworks.However,therstkindofnetworksoftenhashighesttotalofcapacitieswhileithasthesamenodeloadasthe 34

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AWSNnetwork BSynthesizednetwork(d=4) Figure2-5. VulnerabilityofWSNnetworkunderscaledsafetysetting remainedtwonetworks.Thus,wesetuptheexperimenttomeasuretherobustnessofeachkindofnetworksasfollows.Werstgeneratethesafenetwork,thenwechooseasuitableTvaluetogeneratenormalandscaledsafenetworksuchthatthetotalcapacityisthesame.Fig. 2-6 showsthatscaledsafenetworkarethemostrobustone.Normalandsafenetworkshaveverycloserobustnessalthoughthesafefactorcanhelptoavoidtherstattack.Whenmultiplenodesareattacked,N-1failuretolerancesettingdoes 35

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nothelpmuch.Thefailureoftherstnodemakessomeothernodesreachthefailurelimit.Evenatinyadditionalloadcanmakethemfailandtriggeralargecascadingfailure.Therefore,weobservethattheN-1safecriteriondoesnotprotectpowernetworksundercascadingattack. AWSN(T=2,=1) BS4(T=1.6,=1) Figure2-6. Networkrobustnesswithdifferentfailuretoleranceschemes 2.5.4VertexLoadandNetworkRobustnessNowwevarythevalueofandmeasuretherobustnessofnetworksundertheFACPattackingstrategy.AsillustratedinFig. 2-7 ,thehigheris,themorevulnerablethenetworkisastheloadisconcentratedatafewnodes.ThesenodesbecometheAchilles'heelofthenetwork.Itsuggeststhatweshoulddistributetheworkloadofnodesinnetworkstomakethemlessvulnerable. AWSN(T=2,=1) BS4(T=1.6,=1) Figure2-7. Networkrobustnessunderdifferentloaddistribution 36

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2.5.5NetworkTopologyandNetworkRobustnessWenextevaluatetherobustnessofnetworkswithdifferentaveragedegree.AsshowninFig. 2-8 ,thenetworkwithhigheraveragedegreeismorerobustthannetworkwithloweraveragedegree.Inthedensernetwork,theloadofafailednodeisredistributedtoalargernumberofneighbors.Eachneighborhoodnodereceivesasmallfractionoftheload,thusitcanbeartheadditionalloadwithoutfailing. Figure2-8. Networktopologyandrobustness 2.6RelatedWorksThecascadingfailurehasattractedalotofattentionandbeenstudiedinvariousperspective[ 6 22 30 40 46 53 54 60 ].Thestructuralvulnerabilityofpowernetworkswasstudiedin[ 6 ].Theauthorsshowedthatremovingsmallfractionofhighestdegreenodessignicantlyreducestheconnectivityofthenetwork.Afterthat,Hinesetal.[ 30 ]studiedthenetworkvulnerabilityofdifferentclassesofscale-freenetworksincludingErdos-Renyi,preferential-attachment,andsmall-worldnetworks.Theyshowedthatdifferenttypesofnetworksbehavedifferentlyundernodefailures.Therevariousmodelsofcascadingfailureswerelaterproposedtovulnerabilityofnetworksundertargetedattack[ 7 36 53 54 ].However,theseworksmainlypresentdifferentrankingmethodsfornodesandselectmostcriticalnodes.Thesemethodsfailtoaddresstheeffectofthecascadingprocess,hencemisstothemostcriticalnodes. 37

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2.7SummaryInthischapter,westudiedthecriticalnodedetectionproblemunderloadredistributioncascadingmodel.Basedonthenewcentralitywhichconsidersthecascadingeffectwhenevaluatetheimportanceofnodesinnetworks,wedevelopvariousalgorithmstoidentifycriticalnodes:oneispurelybasedonthenewcentrality,oneisbasedonpartiallyadaptivecentrality,andoneisbasedonfullyadaptivecentrality.Amongthem,thefullyadaptivecentralityalgorithmcontinuouslyupdatesthecentralityofnodesandselectthebestone,henceachievesthehighestperformanceamongthethree.However,thisalgorithmsufferswhenthenetworkishighlytoleranttothecascadingfailure.Weproposethecooperatingattackalgorithmwhichcooperatesselectednodestotakedownprotectednodeswithhighcapacity.Theperformanceguaranteeofthecooperatingattackalgorithmissupportedbyboththeoreticalandexperimentalresults.Inaddition,weuseproposedalgorithmstostudythevulnerabilityofdifferentsafetysettings.Wendthatnetworkswithlowdensityisextremelyvulnerableunderthecascadingfailure.Inthiskindofnetworks,theloadofafailednodeisredistributedtoasmallnumberofneighborsandcanfailsthemeasily.Ontheotherhand,theloadisshredtosmallerportionsinnetworkswithhighdensity.Wealsodiscoverthatevenwithnetworkofthesametopologyandtotalcapacityandload,thenetworksafetydependsalotonthedistributionofprotectioncost(thegapbetweenthecapacityandtheload).Theseshredthelightondesigningsafenetworks. 38

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CHAPTER3CASCADINGFAILUREOFNODESININTERDEPENDENTNETWORKSInthedevelopmentoftechnology,infrastructurenetworksaremoreandmoreinterdependentoneachothertooperateproperly.Tooptimizetheoperationperformanceandreducetheeconomicspend,thesenetworkstendtoutilizethesupportfromotherones.AtypicalexampleistheSmartGridinwhichthepowernetworkusesthecommunicationnetworktoexchangeoperationalinformationandthecommunicationnetworkusestheelectricityfromthepowernetworktooperate.Inthemeanwhile,suchgrowinginterdependenciesalsodramaticallyimpactthevulnerabilityofthesenetworkssinceanetworkisnotonlyexposestothreatstothemselvesbutalsotothecascadingfailuresinducedbyfromothernetworks.Inatypicalattackingpointofview,anattackerwouldrstexploitthenetworkweaknesses,andthenonlyneedstotargetonsomecriticalnodesineitherpowernetworksortheirinterdependentcommunicationnetworks,whosecorruptionsbringthewholenetworkdowntoitsknees.Inotherwords,nodesfrompowernetworksdependheavilyonnodesfromtheirinterdependentnetworksandviceversa.Consequently,whennodesfromonenetworkfail,theycausenodesintheothernetworktofail,too.Forinstance,anadversarialattacktoanyessentialInternethosts,e.g.tier-1ISPssuchasQwest,AT&TorSprintservers,oncesuccessful,maycausetremendousbreakdownstobothmillionsofonlineservicesandthefurtherlarge-areablackoutbecauseofthecascadingfailures.Areal-worldexampleisthewide-rangeblackoutthataffectedthemajorityofItalyon28September2003[ 47 ],whichresultedfromthecascadingfailuresinducedbythedependencebetweenpowernetworksandcommunicationnetworks.Therefore,inordertoguaranteetherobustnessofpowernetworkswithoutreducingtheirperformancebydecouplingthemfrominformationsystems,itisimportanttoidentifythosecriticalnodesininterdependentpowernetworks,beforehand. 39

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Figure3-1. ExampleofInterdependentPowerNetworkandCommunicationNetwork Therehavebeenmanystudiesassessingthenetworkvulnerability[ 6 8 15 27 32 45 48 ].Yet,theseapproachesareeitherdesignedonlyforsinglenetworksorheavilydependentoncongurationmodelsofinterdependentnetworks.Theexistingapproaches[ 5 6 8 39 ]forsinglenetworksarebasedonvariousmetrics,suchasthedegreeofsuspectednodesoredges[ 6 ],theaverageshortestpathlength[ 5 ],theglobalclusteringcoefcients[ 39 ],andthepairwiseconductivity[ 6 8 ]andsoon.However,whenapplyingintointerdependentnetworks,theirperformancesdroptremendouslysincethesemetricsfailtocastthecascadingfailuresininterdependentnetworks.Lateron,otherresearchers[ 15 27 32 45 48 ]studiedthevulnerabilityassessmentoninterdependentnetworks,basedonthesizeoflargestconnectedcomponentinpowernetworksaftercascadingfailures.Althoughtheyshowedtheeffectivenessofthisnewmetric,mostofthemfocusonthearticialmodelsofinterdependentnetworks,i.e.,randominterdependencybetweennetworks,andignorethedetectionoftopcriticalnodesinrealnetworks.LetusconsiderasimpleexampleofinterdependentnetworksinFig. 3-1 ,whichillustratesasmallportionofpowernetwork(lowernodes),communicationnetwork(uppernodes)andtheirinterdependencies(dottedlinks).Whenweonlytakethesinglepowernetworkintoaccount,thefailureofu7destructsthepowernetworkmorethanthatofu1sincethelargestconnectedcomponentisofsize6(fu1,u2,...,u6g)whenu7 40

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fails,whichissmallerthan9(fu2,...,u10g)whenu1fails.However,ifconsideringitsinterdependenceuponthecommunicationnetwork,thefailureofu1willdestroythepowernetworkmorethanthatofu7.Thisisbecausethefailureofu1causesthefailureofv1inthecommunicationnetwork,whichfurtherfailsv2,v3,andv4sincetheyaredisconnectedfromthelargestconnectedcomponent.Duetotheirinterdependenceofthenodesv4andu7inthepowernetwork,thesecascadingfailuresnallyresultinthelargestconnectedcomponentofthepowernetworktobefu8,u9,u10gofsizeonly3.Yet,thelargestconnectedcomponentremainsthesameasinasinglepowernetworkafterthefailureofu7.Thisexampleillustratesanimportantpointthattheroleofonenodecouldbetotallydifferentbetweensingleandinterdependentnetworkswithrespecttothevulnerabilityassessment.Inthischapter,weinvestigatethevulnerabilityofinterdependentnetworkswhenthecascadingfailureshappenbasedontheconnectivityofnodes.Whenstudyinginterdependentnetworks,especiallythepowernetworkandthecommunicationnetwork,itiswellknowntoassumethatanodewillfailurewhenitisdisconnectedfromthelargestconnectedcomponent.Althoughtheassumptiondoesnotcapturethereality,itcontainsarealisticmeaning.Inrealworld,whenanodeisdisconnectedthelargestconnectedcomponent,itisalmostremovedfromthesourceofpowerorinformation.Inthischapter,westudytheprobleminthecontextofpowernetworkandcommunicationnetwork,buttheresultsisappliedforallsystemsthatsharethesamecascadingmechanism.Therestofthechapterisorganizedasfollows.InSection 3.1 ,weintroducetheinterdependentnetworkmodelandtheproblemdenition.Afterthat,Section 3.2 includesthehardnessandinapproximabilityresults.ThegreedyframeworkisproposedinSection 3.3 ,alongwiththecentralitymetric.TheexperimentalevaluationisillustratedinSection 3.4 .TherelatedworkispresentedinSection 3.6 .Finally,Section 3.7 providessomeconcludingremarks. 41

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3.1NetworkModelandProblemDenitionInthissection,werstintroduceourinterdependentnetworkmodelandthewell-acceptedmodelofcascadingfailure.Usingthesemodels,westudytheInterde-pendentPowerNetworkDisruptorproblem,tominimizethesizeoflargestconnectedcomponentsinthepowernetworkaftercascadingfailuresbyselectingacertainnumberoftargetnodes. 3.1.1InterdependentNetworkModelConsideringaninterdependentsystem,weabstractitintotwographs,Gs=(Vs,Es)andGc=(Vc,Ec),andtheirinterdependencies,Esc.GsandGcrepresentthepowernetworkandcommunicationnetworkrespectively.EachofthemhasasetofnodesVs,VcandasetoflinksEs,Ec,whicharereferredtoasintra-links.Inaddition,Escareinter-linkscouplingGsandGc,i.e.,Esc=f(u,v)ju2Vs,v2Vcg.Anodeu2VsisfunctionalifitisconnectedtothegiantconnectedcomponentofGsandatleastoneofitsinterdependentnodesinGcisinaworkingstate.ThewholeinterdependentsystemisreferredtoasI(Gs,Gc,Esc). 3.1.2CascadingFailuresModelWeuseawell-acceptedcascadingfailuremodel,whichhasbeenvalidatedandappliedinmanypreviousworks[ 15 27 32 45 48 ].Initially,thereareafewcriticalnodesfailedinnetworkGs,whichdisconnectsasetofnodesfromthelargestconnectedcomponentofGs.Duetotheinterdependencyoftwonetworks,allnodesinGconlyconnectingtofailednodesinGsarealsoimpacted,andthereforestopworking.Furthermore,thefailurescascadetonodeswhicharedisconnectedfromthelargestconnectedcomponentinGcandcausefurtherfailuresbacktoGs.Theprocesscontinuesbackandforthbetweentwonetworksuntilthereisnomorefailurenodes. 3.1.3ProblemDenition Denition3(IPNDproblem). GivenanintegerkandaninterdependentsystemI(Gs,Gc,Esc),whichconsistsoftwonetworksGs=(Vs,Es),Gc=(Vc,Ec)alongwith 42

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theirinterdependenciesEsc.LetLGs(T)bethesizeofthelargestconnectedcomponentofGsafterthecascadingfailurescausedbytheinitialremovalofthesetofnodesTVsinGs.TheIPNDproblemasksforasetTofsizeatmostksuchthatLGs(T)isminimized.Intherestofthechapter,thepairsoftermsinterdependent,networksandcouplednetworks,nodeandvertex,aswellasedgeandlink,areusedinterchangeably. 3.2ComputationalComplexityInthissection,werstshowtheNP-completenessofIPNDproblembyreducingitfrommaximumindependentsetproblem,whichfurtherimpliesthatIPNDproblemisNP-hardtobeapproximatedwithinthefactor2)]TJ /F12 11.955 Tf 11.96 0 Td[("forany">0. Theorem3.1. IPNDproblemisNP-complete. Proof. ConsiderthedecisionofIPNDthataskswhetherthegraphGs=(Vs,Es)inaninterdependentsystemI(Gs,Gc,Esc)containsasetofverticesTVsofsizeksuchthatthelargestconnectedcomponentinGs[VsnT]aftercascadingfailuresisatmostzforagivenpositiveintegerz.GivenT2Vs,wecancomputeinpolynomialtimethesizeofthelargestconnectedcomponentinGsafterthecascadefailureswhenremovingTbyiterativelyidentifyingthelargestconnectedcomponentandremovingdisconnectedverticesinGsandGc.ThisimpliesIPND2NP.ToprovethatIPNDisNP-hard,wereduceitfromthedecisionversionofthemaximumindependentset(MIS)problem,whichasksforasubsetIVwiththemaximumsizesuchthatnotwoverticesinIareadjacent.LetanundirectedgraphG=(V,E)wherejVj=nandapositiveintegerkjVjbeanyinstanceofMIS.NowweconstructtheinterdependentsystemI(Gs,Gc,Esc)asfollows.WeselectGs=GandGctobeacliqueofsizejVsj.SinceGsandGchavethesamesizeinourconstruction,toconstructtheinterdependencybetweenGsandGc,werandomlymatcheachnodeinVstosomearbitrarynodesinVcsoastoformaone-to-onecorrespondencebetweenVsandVc.AnexampleisillustratedinFig. 3-2 .WeshowthatthereisanMISofsize 43

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atmostkinGiffGsinI(Gs,Gc,Esc)hasanIPNDofsizen)]TJ /F3 11.955 Tf 12.68 0 Td[(ksuchthatthelargestconnectedcomponentinGsaftercascadingfailuresisofsizeatmost1. Figure3-2. AnexampleofreductionfromMIStoIPND First,supposeIVisanMISforGwithjIjk.Byourconstruction,thelargestconnectedcomponentofGs[I]hasthesize1sincethereisnomorecascadingfailureinthecliqueGc.Thatis,VsnIisalsoanIPNDofI(Gs,Gc,Esc).Conversely,supposethatT0VswithjT0j=n)]TJ /F3 11.955 Tf 12.48 0 Td[(kisanIPNDofI(Gs,Gc,Esc),thatis,thelargestconnectedcomponentofGs[VsnT0]isofsize1.WeshowthatVsnT0isalsoanMISofG.SincethefailureofnodesinGcwillnotcauseanycascadingfailureinGs,thelargestconnectedcomponentofGs[VsnT0]isofsize1iffVsnT0isanindependentsetinGs.Thatis,VsnT0isalsoanMISofG. AsIPNDisNP-complete,onewillquestionhowtightlywecanapproximatethesolution,leadingtothetheoryofinapproximability.Theinapproximabilityfactorgivesusthelowerboundofnear-optimalsolutionwiththeoreticalperformanceguarantee.Thatsaid,no-onecandesignanapproximationalgorithmwithabetterratiothantheinapproximabilityfactor.Then,weshowthattheabovereductionimpliesthe(2)]TJ /F12 11.955 Tf -418.31 -23.91 Td[(")-inapproximabilityfactorforIPNDinthefollowingcorollary. Corollary1. IPNDproblemisNP-hardtobeapproximatedinto2)]TJ /F12 11.955 Tf 11.96 0 Td[("forany">0. Proof. WeusethereductionintheproofofTheorem 3.1 .Supposethatthereisa(2)]TJ /F12 11.955 Tf 12.44 0 Td[(")-approximationalgorithmAforIPND.ThenAcanreturnthelargestconnectedcomponentinGsofsizelessthan2inourconstructedinstanceiftheoptimalsizeis1. 44

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ThusalgorithmAcanbeappliedtosolveMISonGinpolynomialtimebecausethissizeisintegral.ThiscontradictstotheNP-hardnessofMIS. 3.3GreedyFrameworkforIPNDProblemInthispart,wepresentdifferentalgorithmstodetectthetopcriticalnodesusingthegreedyframework,whichhasbeenillustratedasoneofthemostpopularandeffectiveapproachestosolvehardproblems.Theideaistoiterativelychoosethemostcriticalnode,whoseremovaldegradesthefunctionalityofthenetworkasmuchaspossible.Indetail,weproposethreefollowingdifferentstrategiestoselectacriticalnodeinthesystemateachiteration: 1) Selectanodethatmaximizesthenumberoffailednodesafterthecascadingfailure. 2) Selectanodethatdecreasesthestructuralstrengthofthesystemasmuchaspossible.Thatis,whenthenumberofremovednodesislargeenough,thesystemwillbecomeweak.Therefore,thenumberoffailednodesincreasesconsiderablyundertheeffectofcascadingfailures. 3) Selectanodethatnotonlyincreasesthenumberoffailednodesbutdecreasesthestructuralstrengthaswell.Intherestofthissection,wedescribethreealgorithmscorrespondingtotheabovestrategies. 3.3.1MaximumCascading(Max-Cas)AlgorithmInmaximumcascading(Max-Cas)algorithm,weiterativelyselectanodeuthatleadstothemostnumberofnewfailednodes,i.e.,themaximummarginalgaintothecurrentsetofattackednodesT.Whenanewnodeufails,itresultsinachainofcascadingfailures.Thenumberofnewfailednodes,referredtoascascadingimpactnumber,canbecomputedbysimulatingthecascadingfailureswiththeinitialsetT[fugontheinterdependentsystemIasdescribedinSection 3.1.2 .However,thesimulationofcascadingfailuresistime-consumingduetoitscalculationofcascadingfailures 45

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betweentwonetworks.Eachstepinthecascadingrequirestoidentifythelargestconnectedcomponentofeachnetwork.Tothisend,wefurtherimprovetherunningtimeofouralgorithmbyreducingthenumberofsimulations.Theideaisonlytocheckpotentialnodeswhoseremovalcreatesatleastonemorefailednodeinthesamenetworkduetothecascadingfailures.Thatis,thisnode(oritscouplednode)disconnectsthenetworkwhichitbelongsto,i.e.,it(itscouplednode)isanarticulationnodeofGs(orGc),whichisdenedasanyvertexwhoseremovalincreasesthenumberofconnectedcomponentsinGs(orGc).Thereasonisillustratedinthefollowinglemma. Lemma1. GivenaninterdependentsystemI(Gs,Gc,Esc),removinganodeu2Vsfromthesystemcausesatleastonemorenodefailduetothecascadingfailureiffu(oritscouplednodev2Vc)isanarticulationnodeinGs(orGc). Proof. IfuisanarticulationnodeofGs,theremovalofuwillincreasethenumberofconnectedcomponentsinGsatleasttotwo.BythedenitionofcascadingfailuresinGs,allnodesdisconnectedfromthelargestconnectedcomponentwillbefailed.Similarly,whenvisanarticulationnodeofGc,removingucausesvfail,thenthereisatleastonemorenodesinGcfail.Afterthat,thesenodesmakecouplednodesinGsfailaswell.Ontheotherhand,ifneitherunorvarearticulationnodes,theremovalofuonlymakesvfail,andtherestoftwonetworksarestillconnected,whichterminatesthecascadingfailures. Accordingtothisproperty,theproposedalgorithmrstidentiesallarticulationnodesinbothresidualnetworksusingHopcroftandTarjan'salgorithm[ 31 ].Notethatthisalgorithmrunsinlineartimeonundirectedgraphs,whichisfasterthanonesimulationofcascadingfailures.Thus,therunningtimeofeachiterationissignicantlyimprovedespeciallywhenthenumberofarticulationnodesissmall.DenoteMax)]TJ /F3 11.955 Tf -429.87 -23.91 Td[(Cas(Gs,T,fug)astheimpactnumberofu,Algorithm 5 describesthedetailstodetectcriticalnodes.InAlgorithm 5 ,sinceittakesO(n)timetocomputethecascadingimpact 46

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numberforeachnodeandatmostjAj
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Intuitively,thisnewcentralitymeasureisrequiredtocaptureboththeintra-centrality(thecentralityofnodesineachnetworks)andinter-centrality(thecentralityformedbytheinterconnectionsbetweentwonetworks).GivenaninterdependentsystemI(Gs,Gc,Esc),nodeuinVsismorelikelytobecriticalifitscouplednodev2Gciscritical.Furthermore,whennodeuisconsideredasacriticalnode,itsneighborsarealsomorelikelytobecomeimportantsincethefailuresofthesenodescancauseufail.Thatsaid,thecriticalityofthesenodesimplythecriticalityoftheircouplednodes.Tocapturethiscomplicatedrelationininterdependentsystems,wedevelopaniterativemethodtocomputethecentralityofnodes,calledIterativeInterdependentCentrality(IIC).Initially,thecentralitiesofallnodesinGsarecomputedbythetraditionalcentrality,e.g.,degreecentrality,betweennesscentrality,etc.Afterthat,thesecentralitiesofnodesinGsarereectedtocouplednodesinGcandthecentralitiesofnodesinGcareupdatedbasedonthereectedvalues.ThecentralitiesofnodesinGccontinuetobereectedonnodesofGsandupdatecentralitiesofthesenodes.TwokeypointsofIICaretheupdatingfunctionandtheconvergence. 3.3.2.1UpdatingfunctionConsideringthereectedvaluesfromtheothernetworkastheweightofnodes,wemodifytheweighteddegreeastheupdatedcentralityofnodes,whichisdenedasC(u)=w(u)+(1)]TJ /F12 11.955 Tf 11.96 0 Td[()Xv:(u,v)2Ew(v) dvwherew()isthereectedvalues(ortheweightofnodes)andthereservationfactorlyingintheinterval[0,1].Theunderlyingreasonweuseweighteddegreeisthatanodeisusuallymorecriticalifmostofitsneighborsarecriticalnodes.Thereservationfactorshowsthattheimportanceofeachnodeisnotonlydependentonthereectedvaluesfromtheothernetwork,butalsotheroleinitsownnetwork. 48

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3.3.2.2ConvergenceNext,weshowthatthecentralitiesofnodescanbecomputedbasedonmatrixmultiplicationsandprovetheconvergenceviathisproperty.Letxt=[xtvs1,xtvs2,...,xtvsn],yt=[ytvc1,ytvc2,...,ytvcn]bethenormalizedcentralityvectoraftertthiterationofGsandGc.Supposethattwointerdependentnodeshavethesamepositionvectorsxtandyt,i.e.,vsiandvciareinterdependent.Then,wehavextu=yt)]TJ /F9 7.97 Tf 6.59 0 Td[(1u+(1)]TJ /F12 11.955 Tf 11.95 0 Td[()Xv:(u,v)2Esyt)]TJ /F9 7.97 Tf 6.59 0 Td[(1v dv,8u2Vsytu=xt)]TJ /F9 7.97 Tf 6.59 0 Td[(1u+(1)]TJ /F12 11.955 Tf 11.96 0 Td[()Xv:(u,v)2Ecxt)]TJ /F9 7.97 Tf 6.59 0 Td[(1v dv,8u2VcNotethatifwedividethesevectorsbyaconstant,thentheystillrepresentthecentralityofnodesinthesystems.Thus,aftereachiteration,thesecentralityvectorsaredividedbysomeconstantsCsandCcwhicharechosenlatertoprovetheconvergence.xt=xt=Cs,yt=yt=CcLetMsandMcbennmatricessuchthatMsuv=8>>>><>>>>:ifu=vd)]TJ /F9 7.97 Tf 6.58 0 Td[(1vif(u,v)2Es0otherwiseMcuv=8>>>><>>>>:ifu=vd)]TJ /F9 7.97 Tf 6.59 0 Td[(1vif(u,v)2Ec0otherwiseThentherelationshipbetweenxtandytisrewrittenas:xt=Msyt)]TJ /F9 7.97 Tf 6.59 0 Td[(1 Cs,yt=Mcxt)]TJ /F9 7.97 Tf 6.59 0 Td[(1 Cc 49

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Thereforext=MsMcxt)]TJ /F9 7.97 Tf 6.59 0 Td[(2 CsCc=Mxt)]TJ /F9 7.97 Tf 6.58 0 Td[(2 CsCcwhereM=MsMciscalledthecharacteristicmatrix.Next,weanalyzetheconditionofthismatrixtoguaranteethatxtconvergesbyusingtheJordancanonicalformofM,denedasfollows. Theorem3.2(JordanCanonicalForm[ 50 ]). AnynnmatrixMwithneigenvaluesj1jj2j...jnjcanberepresentedasM=PJP)]TJ /F9 7.97 Tf 6.59 0 Td[(1wherePisaninvertiblematrixandJisJordanmatrixwhichhasformJ=diag(J1,...,JP)whereeachblockJi,calledJordanblock,isasquarematrixoftheformJi=266666664i1i......1i377777775Accordingtoitsabovedenition,thepowerofthematrixMcanbecomputedasfollowsMk=(PJP)]TJ /F9 7.97 Tf 6.59 0 Td[(1)k=PJkP)]TJ /F9 7.97 Tf 6.59 0 Td[(1Hence,Mkconvergeswhenk!1ifandonlyifJkconverges.ThepowersofJiscomputedviathepowersofJordanblockJk1,Jk2,...,Jkp.Jk=diag(Jk1,...,JkP) 50

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whereJki=266666664ki)]TJ /F8 7.97 Tf 5.48 -4.38 Td[(k1k)]TJ /F9 7.97 Tf 6.59 0 Td[(1i)]TJ /F8 7.97 Tf 5.48 -4.38 Td[(k2k)]TJ /F9 7.97 Tf 6.59 0 Td[(2i)]TJ /F8 7.97 Tf 12.01 -4.38 Td[(kdi)]TJ /F9 7.97 Tf 6.59 0 Td[(1k)]TJ /F9 7.97 Tf 6.59 0 Td[((di)]TJ /F9 7.97 Tf 6.59 0 Td[(1)iki)]TJ /F8 7.97 Tf 5.48 -4.38 Td[(k1k)]TJ /F9 7.97 Tf 6.59 0 Td[(1i)]TJ /F8 7.97 Tf 12.01 -4.38 Td[(kdi)]TJ /F9 7.97 Tf 6.59 0 Td[(2k)]TJ /F9 7.97 Tf 6.59 0 Td[((di)]TJ /F9 7.97 Tf 6.59 0 Td[(2)i...ki377777775NotethatthepowersofJconvergesifandonlyifthepowersofallJordanblocksconverge.Thus,wefocusontheconvergenceofablockJkasstatedinthefollowinglemma. Lemma2. TheconvergenceofaddJordanblockJidependsonlydandi:(1)Ifjij>1thenJkidoesnotconvergewhenk!1.(2)Ifjij<1thenJkiconvergesto0whenk!1.(3)Ifjij=1andi6=1,thenJkidoesnotconvergewhenk!1.(4)Ifi=1andd=1,thenJki=1.(5)Ifi=1andd>1,thenJkidoesnotconvergewhenk!1. Proof. Cases(1),(3),(4),and(5)aretrivial,thusweonlyshowtheproofforcase(ii).Withjij<1,everyelementofJkihasform)]TJ /F8 7.97 Tf 5.48 -4.38 Td[(kjk)]TJ /F8 7.97 Tf 6.59 0 Td[(jiwhichconvergeto0ask!1. Accordingtothislemma,whennormalizedfactorsCs,CcsatisesCsCc=j1j,wewillhaveM CsCct=2x0=PJ j1jt=2P)]TJ /F9 7.97 Tf 6.59 0 Td[(1x0Clearly,xtwillconvergewhenJ j1jt=2converges.Then,wehavethefollowingtheorem. Theorem3.3. Thecentralityvectorconvergesifandonlyifthecharacteristicmatrixhasexactlyonemaximummagnitudeeigenvalue.Tocomputetheconvergedcentralityvector,werstchoosesuchthatMhas1>2.Inpractice,wechoose=0.5andcentralityvectorsalwaysconverge. 51

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AlthoughitseemsnecessarytocomputethelargesteigenvalueofM,weproposeanalternativemethodtoavoidthistime-consumingcomputationasfollows.Supposethatx2tconvergestoavectorxaftert0iterationsi.e.x=Mt0x0 j1jt0.Nowwedenethesequenceofvectorsz0=x0,zi+1=Mzi jMzij,then:zt0=Mt0z0 Qt0)]TJ /F9 7.97 Tf 6.58 0 Td[(1i=0jMzij=Mt0x0 Qt0)]TJ /F9 7.97 Tf 6.59 0 Td[(1i=0jMzijItmeansthatx=Azt0whereAisascalarvalue.Thereforezt0=x jjxjj.ThuswecancomputethecentralityvectorusingtherecursiveformulaofzasdescribedinAlgorithm 6 ,thenusethisalgorithmassub-routinetodetectcriticalnodesinAlgorithm 7 Algorithm6IterativeInterdependentCentrality Input:CharacterizematrixMandallowederror Output:Centralityvector x 1 error +1 whileerror>do y Mx norm jjyjj y y=norm error jjy)]TJ /F18 11.955 Tf 11.95 0 Td[(xjj x y endwhile Returnx Algorithm7IIC-basedAlgorithm Input:InterdependentsystemI(Gs,Gc,Esc),anintegerk Output:SetofkcriticalnodesinT2Vs T ; fori=1tokdo 0.5,ComputeM 10)]TJ /F9 7.97 Tf 6.59 0 Td[(8 ComputecentralityvectorxusingAlgorithm 6 u argmaxVsnTx[u] T T[fug UpdateI[VsnT] endfor ReturnT 52

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TimeComplexity:SincetwomatricesMsandMchaveonly(2jEsj+jVsj)and(jEcj+jVcj)non-zeroelements,theproductMx=MsMcxtakesO(2jEsj+jVsj+2jEcj+jVcj)timeusingsparsematrixmultiplication.Theconvergencespeedisj1j j2j,thusthenumberofiterationsisO(log(1=) logj1j j2j).Therefore,thetotalrunningtimetocomputeiterativeinterdependentcentralityisO((jEs+Ecj)log(1=) logj1j j2j).Thus,thetotaltimetodetectcriticalnodesisO(k(jEs+Ecj)log(1=) logj1j j2j). 3.3.3HybridAlgorithmMotivatedbytheadvantagesofMax-CasandIICalgorithms,wefurtherdesignahybridalgorithmbytakingadvantageofthem.Asonecansee,Max-Casonlyworkswellwhennetworksarelooselyconnectedsinceitmainlyaimstocreateasmanyfailednodesaspossibleinsteadofmakingthenetworkasweakaspossible.Ontheotherhand,IICalgorithmcanmakethenetworkweakbutitdoesnotworkwellasMax-Caswhennetworksarelooselyconnected.Thus,theideaofhybridalgorithmistoremoveasmanynodesaspossibleandmakenetworksweakerinturn.Thatis,weuseMax-CasandIICalgorithmsinoddandeveniterationsrespectively,asdescribedinAlgorithm 8 .SincetherunningtimeofIICismuchsmallerthanMax-Cas,itsrunningtimeisaboutahalfofMax-Cas,whichwillbeempiricallyshowninSection 3.4 Algorithm8HybridAlgorithm Input:InterdependentsystemI(Gs,Gc,Esc),anintegerk Output:SetofkcriticalnodesinT2Vs T ; fori=1tokdo ifiisoddthen SelectuasMax-Casalgorithm else SelectuasIICalgorithm endif T T[fug UpdateI[VsnT] endfor ReturnT 53

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3.4ExperimentalEvaluation 3.4.1DatasetandMetricIntheexperiment,weevaluateMax-Cas,IIC,andHybridalgorithms,withrespecttothesizeofgiantconnectedcomponent(GCC)andtherunningtime,onvariousreal-worldandsyntheticdatasets.Intermsofpowernetworks,weusebothrealWesternStatespowernetworkoftheUS[ 55 ]with4941nodesand6594edges,andthesyntheticscalefreenetworks.Thisnetworkaswellasothercommunicationnetworksbelongtoaclassofnetworkscalledscale-freenetworksinwhichthenumberofnodeswithdegreed,denotedbyP(d),isproportionaltod)]TJ /F11 7.97 Tf 6.59 0 Td[(i.e.,P(d)d)]TJ /F11 7.97 Tf 6.59 0 Td[(forsomeexponentialfactor>0.Accordingto[ 9 ],powernetworksarefoundtohavetheirexponentialfactorsbetween2.5and4.Inordertodoamorecomprehensiveexperiment,wefurthergeneratemoretypesofsyntheticpowernetworkswithdifferentexponentialfactors,usingigraphpackage[ 23 ].Inaddition,duetothelackofdatadescribinginterdependenciesbetweenanycommunicationnetworksandthereal-worldpowernetwork,weusethesyntheticscale-freenetworks,representingcommunicationnetworks,e.g.Internet,telephonenetwork,etc.Sincemostcommunicationnetworksareobservedtohavethescalefreepropertywiththeirexponentialfactorsbetween2and2.6[ 12 57 ],wegeneratecommunicationnetworkswithcomponentfactorsof2.2or2.6.Forthesakeofcouplingmethod,motivatedbytheobservationfromreal-worldinterdependentsystemsin[ 44 ],wedeveloparealisticandpracticalcouplingapproach,RandomPositiveDegreeCorrelationCoupling(RPDCC)scheme.Inthisscheme,nodeswithhighdegreestendtocoupledtogetherandsodonodeswithlowdegrees,thusthedegreecorrelationofcouplednodesispositiveasdescribedin[ 44 ].ThedetailofRPDCCstrategywillbediscussedlaterinSection 3.5 .Finally,eachexperimentonsynthesizedsystemsisrepeated100timestocomputetheaverageresults. 54

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3.4.2PerformanceofProposedAlgorithmsInordertoshowtheeffectivenessofourproposedalgorithms,duetotheintractabilityofIPNDproblemandthetimeconsumptiontoobtainoptimalsolutions,wefocusoncomparingthemwithtraditionalcentralityapproacheswhichareoftenusedinnetworkanalysis[ 11 ],includingdegreecentrality(DC),closenesscentrality(CC),betweennesscentrality(BC)[ 13 ],andbridgenesscentrality(BRC)[ 33 ].Intheseapproaches,theknodesoflargestcentralitiesinpowernetworksareselectedascriticalnodes.Particularly,wetestourapproachesonthefollowingthreetypesofdatasets: 1) WSSystem:USWesternstatespowernetworkScale-freecommunicationnetworkwith=2.2. 2) SSSystem:Scale-freepowernetworkswith=3.0Scale-freecommunicationnetworkwith=2.2. 3) Eq-SSSystem:Scale-freepowerandcommunicationnetworkswiththesame=2.6.Herewechoosetheexponentialfactoraccordingtothereal-worldpowernetworksandcommunicationnetworks,asmentionedabovein 3.4.1 .Fig. 3-3 reportsthecomparisonofperformancebetweendifferentapproachesinthesethreeinterdependentsystems.Inthesegures,allofthreeproposedalgorithmsoutperformCC(thebesttraditionalcentralityapproach)foranynumberofkcriticalnodes.Whenkbecomeslarger,theinterdependentsystemshavetotallydestroyedbychoosingthesecriticalnodesusingouralgorithms,whilemorethan60%ofnodesremainintactifselectingnodeswithhighesttraditionalcentralities.EspeciallyinWSinterdependentsystemconsistingofreal-worldUSWesternstatespowersystem,thenumberoffunctionalnodesremainsnearly5000even50nodesareidentiedusingCC,whereasitissufcienttodestroythewholesystemonlybyremoving20nodesusingourHybridorMax-Casapproach.Thatis,thesetraditionalapproachesperformmuchworsecomparedwithouralgorithms,especiallywhenthenumberofattackednodesislarge.Thus,thetraditionalmethodscannotidentifyacorrectsetofcriticalnodes 55

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ininterdependentsystems.Thereasonisthattheseapproachescanonlyreecttheimportanceofeachnodeinsinglepowernetworksratherthaninterdependentsystems,andtheyignoretheimpactofcascadingfailurestointerdependentsystems. A B C D E F Figure3-3. PerformanceComparisononDifferentInterdependentSystems:WSSystem(A,B),SSSystem(C,D),andEq-SSSystem(E,F). Comparingourthreeproposedapproaches,asrevealedinFig. 3-3 ,IICrunsfastestinspiteofitsworstperformance,roughly1000timesfasterthanMax-CasinWSinterdependentsystem.WealsonoticethattheperformanceofMax-CasandHybrid 56

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algorithmsisveryclosewhileHybridalgorithmrunsabout2timesfasterthanMax-Casalgorithm.Inparticular,Max-CashasbetterperformancethanHybridalgorithminSSinterdependentsystem,yetworseperformanceintheothertwosystems.Thisisbecausethepowernetworkwith=3.0isverylooselyconnectedandfragileinSSinterdependentsystems.Thus,Max-Casstrategycandestroythesystemquicklyandeasily.However,sincenodesarebetterconnectedintheothertwosystems,especiallyEq-SS,Hybridalgorithmismoreefcientduetoitsstrategythatmakesnetworksweakrstandthendestroysthem.AsillustratedinFig. 3-3E ,theperformanceofHybridislowerthanMax-Casinitially,buthigherthanMax-Caswhenthenetworksgetweakenough.Additionally,inallofthesesystems,whenthenumberofremovednodesreachtoacertainvalue,thewholesystemisfailed.Thesenumbersareabout20forWSandSSsystemand40forEq-SSsystem.Thisshowsthatinterdependentnetworksarevulnerable,especiallywhenlooselyconnected. 3.4.3VulnerabilityAssessmentofInterdependentSystemsWiththeeffectivenessofHybridalgorithmobservedthroughtheaboveexperiments,wecondentlyuseittofurtherassessthevulnerabilityofinterdependentsystemsandexploresomeinsightproperties. 3.4.3.1DifferentcoupledcommunicationnetworksWeareinterestedininvestigatingthevulnerabilityofacertainpowernetworkwhenitiscoupledwithdifferentcommunicationnetworks.First,wexonesyntheticpowernetworkbygeneratingascale-freenetworkwith=3accordingto[ 9 ].Thecoupledcommunicationnetworksarealsogeneratedasscale-freenetworks,withtheirexponentialfactorsbetween2.5and2.7,asmentionedabove.Allgeneratednetworkshave1000nodes.AsillustratedinFig. 3-4 ,thepowernetworkstendtobemorevulnerablewhentheircoupledcommunicationnetworksaremoresparse,i.e.,withlargerexponentialfactor.Thatis,itgivesusanintuitionthatthepowernetworkswillbecomemorevulnerable 57

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Figure3-4. TheVulnerabilityOfAFixedPowerNetwork whentheircouplednetworksareeasytobeattacked.Inparticular,inordertodestroythepowernetworks,thenumbersofcriticalnodesinthemare23,17,and11whentheircoupledcommunicationnetworkshave=2.5,=2.6and=2.7,respectively,whichindicatessomekeythresholdstoprotectthefunctionofpowernetworkswiththeknowledgeoftheirinterdependentnetworks. 3.4.3.2DisruptorthresholdInthispart,weevaluateanimportantindicatorofthevulnerability,thedisruptorthresholdwhichisthenumberofnodeswhoseremovaltotallydestroysthewholesystem.Thesmalleritis,themorevulnerablethesystemis.Wewouldliketoobservethedependenceofthedisruptorthresholdonthenetworksize.Particularly,wegeneratetwoscale-freenetworkswiththesamesizeandexponentialfactorsof3.0and2.2,correspondingtopowerandcommunicationnetworks,thencouplethemusingRPDCCscheme.AsshowninFig. 3-5 ,thedisruptorthresholdprovidedbyallproposedalgorithmsissmallandincreasesslowlywithrespecttothegrowthofthenetworksize.Whenthenetworksizeisraisedby5times,from1000to5000nodes,thedisruptorthresholdonlyincreasesroughly3times.Whenthesizeofnetworkis5000,thedisruptor 58

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Figure3-5. TheDisruptorThresholdwithDifferentNetworkSizes thresholdsofMax-CasandHybridalgorithmsareroughly51and57.Thisimpliesthattheremovalof1%numberofnodesisenoughtodestroythewholesystem.EventheIICalgorithmneedstodestroyonly1.5%fractionofnodestobreakthesystemdown.Largeinterdependentsystemsseemtobeextremelyvulnerableunderdifferentattackstrategiesduetothefollowingreason.Whenthenetworksizegrowsup,thepossibilitythatahighdegreenodeisdependentonalowdegreenodealsorunsup.Asaresult,itiseasiertodisablethefunctionalityofhighdegreenodeswhichoftenplayanimportantroleinthenetworkconnectivity.Therefore,thevulnerabilityoftheinterdependentsystemneedstobereevaluatedregularly,especiallyfastgrowingupsystems. 3.4.3.3DifferentcouplingschemesAnotherinterestingobservationistoinvestigatetheimpactsofthewaynodesarecoupledtothevulnerabilityofinterdependentsystem.ApartfromtheRPDCCscheme,weevaluatetherobustnesswithotherthreecouplingstrategies,asfollows: 1) SameDegreeOrderCoupling(SDOC):Thenodesofithhighestdegreeintwonetworksarecoupledtogether. 2) ReversedDegreeOrderCoupling(RDOC):Thenodeofithhighestdegreeinonenetworkiscoupledwiththenodeofithlowestdegreeintheothernetwork. 59

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3) RandomNegativeDegreeCorrelationCoupling(RNDCC):Anodeofhigherdegreeinonenetworkarerandomly,withlowerprobability,tocouplewithanothernodeofhigherdegreenodesintheothernetwork.NotethattheRNDCCschemeistheoppositestrategytotheRPDCCscheme(inSection 3.5 ).Wetestontheinterdependentsystems,consistingofapowernetworkwith=3andacommunicationnetworkwith=2.2usingthefourdifferentcouplingschemes.Allnetworkshave1000nodes.Fig. 3-6 reportsthevulnerabilityofpowernetworkswhencouplingthemwithcommunicationnetworksindifferentmanners.Asonecansee,SDOCprovidesthemostrobustinterdependentsystem,althoughitisnotpractical.Thesizeoftheremainedgiantconnectedcomponentdecreasesslowlywhenthenumberofremovednodesincreases.Ontheotherhand,RDOCmakesthesystemveryvulnerable,whichcanbedestroyedbyonlyremoving1nodesfromthepowernetwork.Thisisbecausethenodesoflowerdegreeincommunicationnetworksareveryeasytobefailed,which,immediately,causethefailurestotheircouplednodesofhigherdegreeinpowernetworks.Whenmanyhighdegreenodesareremoved,thenetworkiseasytobefragmentedwhichleadstothedestructionofthewholesystemshortly.Theinterdependentsystemswiththeothertwoschemes,RPDCCandRNDCC,illustratetheirrobustnessbetweenthoseusingSDOCandRDOC,duetotherandomfactorsinRPDCCandRNDCC.ComparedwithRNDCC,systemscoupledbyRPDCCisalmosttwicemorerobustbecauseofitspositivecorrelations.Theseresultspointoutanimportantprinciplethatthehighercorrelationbetweenthedegreesofcouplednodes,thestrongertheinterdependentsystemis.Inotherwords,anodeofhighdegreeinonenetworkshouldnotbecoupledwithanodeoflowdegreeintheothernetwork;otherwise,thisnodewillbeaweakpointtoattack. 3.5RPDCC/RNDCCCouplingSchemesInthissection,wepresenttheRPDCCschemetorandomlycoupletwonetworkswithpositivedegreecorrelation.GiventwonetworkGsandGc,weformtwoweighted 60

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Figure3-6. VulnerabilityComparisonusingDifferentCouplingSchemes Algorithm9Randomweightedpermutation Input:AweightedsetofnelementsX=fx1,x2,...,xngwithweightsw() Output:WeightedrandompermutationYofX. total Pni=1w(xi) fori=1tondo e arandomselectedelementinXwithprobabilityw(e)=total Y[i] e;X Xnfeg;total total)]TJ /F3 11.955 Tf 11.95 0 Td[(w(e) endfor ReturnY setsthatcontainverticesofGsandGcaselementsandtheirdegreesasweights.Thenwegeneratetworandomweightedpermutationsfvs01,vs02,...,vs0ngandfvc01,vc02,...,vc0ngofnodesinGsandGcasdescribedininAlgorithm 9 ,thenvs0iiscoupledwithvc0i,1in.Inthefollowingtheorem,weshowthatanodeoflargerweighthassmallerexpectedindexineachpermutation,thatis,nodesofhighdegreesintwopermutationstendtohavelowindices.Inotherwords,thisresultsinthepositivecorrelationbetweendegreesofcouplednodes.(ForRNDCC,wecouplevs0iwithvc0n)]TJ /F8 7.97 Tf 6.59 0 Td[(i.) Theorem3.4. Intherandomweightedpermutation,anelementwithbiggerweighthaslowerexpectedindexthananelementwithsmallerweight. 61

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Proof. LetE(X,e)betheexpectedindexofanelementeintherandomweightedpermutation.Then,wehave:E(X,e)=w(e) Px2Xw(x)+Xz2Xnfegw(z) Px2Xw(x)(1+E(Xnfzg,e)Therefore,E(X,e1)E(X,e2)ifw(e1)w(e2). 3.6RelatedWorksMostofworksonnetworkvulnerabilityassessmentarestudiedinsinglenetworks[ 8 10 21 29 41 ].Thecentralitymeasurements[ 11 ]arewidelyused,includingdegree,betweennessandclosenesscentralities,averageshortestpathlength[ 5 ],globalclusteringcoefcients[ 39 ].Alternatively,Arulselvanetal.[ 8 ]rstproposedthetotalpairwiseconnectivityasaneffectivemeasurement,basedonwhichtheyproposetheCNDproblemanddesignedaheuristictodetectcriticalnodes.The-disruptorproblemwaslaterdenedbyDinhetal.[ 25 ]followedbypseudo-approximationalgorithms.Unfortunately,theseapproachesfailtoaccuratelyidentifythecriticalnodesoninterdependentnetworks.Recently,vulnerabilityassessmentofinterdependentnetworkswasinitiatedbyBuldyrevetal.[ 15 ],andfollowedbyasetofrelatedpapers[ 15 27 32 45 48 ].Theseworksvalidatedthesizeoflargestconnectedcomponentasaneffectivemetricforcascadingfailures,coveringawiderangeoftherandomfailures[ 15 ],orderpercolationphasetransition[ 16 27 45 ]andexploitationofrobustnessundertargetedattacks[ 32 ].Theirresultsillustratedthatinterdependentnetworksaremuchmorevulnerablethansinglenetworks.Unfortunately,theseworksheavilydependoncongurationmodelsand,therefore,notapplicabletoreal-worldnetworks.Andnoneofthemproposedastrategytoidentifytopcriticalnodesininterdependentnetworks. 3.7SummaryInthischapter,westudiedtheoptimizationproblemofdetectingcriticalnodestoassessthevulnerabilityofinterdependentpowernetworksbasedonthewell-accepted 62

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cascadingfailuremodelandmetric,thesizeoflargestconnectedcomponent.WeshoweditsNP-hardness,alongwithitsinapproximability.Duetoitsintractability,weproposedagreedyframeworkwithvariousnovelcentralities,whichmeasurestheimportanceofeachnodemoreaccuratelyoninterdependentnetworks.Theextensiveexperimentnotonlyillustratestheeffectivenessofourapproachesinnetworkswithdifferenttopologiesandinterdependencies,butalsorevealsseveralimportantobservationsoninterdependentpowernetworks. 63

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CHAPTER4INFLUENCEDIFFUSIONINMULTIPLEONLINESOCIALNETWORKSIntherecentdecadethepopularityofonlinesocialnetworks,suchasFacebook,Google+,MyspaceandTwitteretc.,hascreatedanewmajorcommunicationmediumandformedapromisinglandscapeforinformationsharinganddiscovery.Onaverage,Facebookusersspend7h:45perpersonpermonth[ 4 ];3.2billionlikesandcommentsarepostedeverydayonFacebook[ 3 ];340milliontweetsaresentouteverydayonTwitter[ 4 ].Suchengagementofonlineusersfertilizesthelandforinformationpropagationtoadegreeneverachievedbeforeinmassmedia.Moreimportantly,OSNsalsoinheritoneofthemajorpropertiesofrealsocialnetworks-theword-of-mouthorpeer-pressureeffectinwhichanindividual'sopinionordecisionisinuencedbyhisfriendsandcolleagues.Duetotheconsiderableimpactofthiseffectonthepopularityofnewproducts[ 14 28 ],OSNshaverapidlybecomeoneofthemostattractivechoicestorisingtheawarenessofnewproductsorbrandsaswellastoreinforcetheconnectionbetweencustomersandcompanies.Thecrucialproblemishowtondthesmallestsetofinuencerswhocaninuenceamassivenumberofusers.ThereisaconsiderablenumberofoverlappingusersamongmultipleOSNswhichcreatesahugeeffectonthediffusionofinformationinthesenetworks.Whenauserjoinsmultiplenetworks,s/hecanrelaytheinformationfromonenetworktoanother.Letusconsiderthefollowingtypicalscenariotoillustratethisphenomenon.Jack,auserofbothTwitterandFacebook,logsinTwitterandknowsaboutanexcellentproductfromhisfriend.Herightawayfallsinlovewiththenewproductandeagerlysharestheinformationbytweetingit.Moreover,heconguredhisTwitterandFacebookaccountsasillustratedinFig. 4-1 thatallowshimtoautomaticallypostonhisFacebook'swallwheneverhehasanewtweetandviceversa.Astheconsequence,theproductinformationisexposedtohisfriendsinbothnetworksandtheinformationfurtherspreadsoutonboththenetworks.Ifweonlyconsidertheinformationpropagationinone 64

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network,thereachoftheinformationisestimatedincorrectlyandthustheinuenceofusersinthesenetworks.Inthiscase,theinuenceofJackshouldbethecombinationofhisinuenceinbothnetworks.AsshowninFig. 4-2 ,thefractionofoverlappingusersisconsiderable,thereforestudyingtheproblemonlyinonenetworkprovidesasolutionwhichisquitedifferentfromreality.ThisprovidesthemotivationtostudytheaboveproblemonmultiplenetworkswheretheinuenceofusersisevaluatedbasedonmultipleOSNsinwhichtheyparticipate.Nearlyalltheexistingworksstudieddifferentvariantsofthemassiveinuenceproblemonasinglenetwork[ 18 19 34 35 37 52 61 62 ].Kempeetal.[ 34 ]rstformulatedtheinuencemaximizationproblemwhichaskstondasetofkuserswhocanmaximizetheinuence.TheinuenceispropagatedbasedonastochasticprocesscalledIndependentCascadeModel(IC)inwhichauserwillinuencehisfriendswithprobabilityproportionaltothestrengthoftheirfriendship.TheauthorprovedthattheproblemisNP-hardandproposedagreedyalgorithmwithapproximationratioof(1)]TJ /F5 11.955 Tf 12.22 0 Td[(1=e).Afterthat,aconsiderablenumberofworksstudyanddesignnewalgorithmsfortheproblemvariantsonthesameorextendedmodelssuchas[ 18 35 ].Therearealsoworksonthelinearthreshold(LT)modelforinuencepropagationinwhichauserwilladoptthenewproductwhenthetotalinuenceofhisfriendssurpasssomethreshold.Fengetal.[ 62 ]showedNP-completenessfortheproblemandDinhetal.[ 24 ]provedtheinapproximabilityaswellasproposedefcientalgorithmsforthisproblemonaspecialcaseofLTmodel.Intheirmodel,theinuencebetweenusersareuniformandauserisinuencedifacertainfractionofhisfriendsareactive.Recently,researchershavestartedtoexploremultiplenetworkswithworksofYaganetal.[ 58 ]andLiuetal.[ 38 ]whichstudytheconnectionbetweenofineandonlinenetworks.TherstworkinvestigatestheoutbreakofinformationusingtheSIRmodelonrandomnetworks.Thesecondoneanalyzesnetworksformedbyonlineinteractionsandofineevents.Theauthorsfocusonunderstandingtheowofinformationand 65

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AAutopostfromFacebooktoTwitter BAutopostfromTwittertoFacebook Figure4-1. Autoupdateacrosssocialnetworks Figure4-2. ThenumberofsharedusersbetweenmajorOSNsin2009[ 2 ] networkclusteringbutnotsolvingthemassiveinuenceproblem.Buttheseworksdonotstudyanyspecicoptimizationproblemofviralmarketing.Shenetal.[ 49 ]exploretheinformationpropagationonmultipleonlinesocialnetworkstakingintoaccounttheinterestandengagementofusers.Intheirsolution,allnetworksarecombinedintoonenetworkbyrepresentinganoverlappinguserasasupernode.Thismethodcannotpreservetheindividualnetworks'properties.Inthischapter,westudytheMassiveInuenceproblem(MIP)whichasksforasetofuserswithminimumcardinalitytoinuenceacertainfractionofusersinmultiplenetworks.Supposethatweknowtheparticipantofusersoverallnetworks,weexploittheadditionalinformationofoverlappinguserstoidentifytopinuentonesovermultiple 66

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networks.Althoughtheproblemhasbeenstudiedinasinglenetworkbutthoseareonlyspecialcasessuchasuniforminuencebetweenusersin[ 24 ],etc.Inaddition,theoverlappingusersintroduceseveralnewchallenges,sotheprevioussolutionscannotbeeasilyadopted.Forexample,howtoevaluatetheinuenceofoverlappingusersacrossmultiplenetworks?Inwhichnetwork,auseriseasiertobeinuenced?Weintroducenovelcouplingschemeswhichcombinemultiplenetworksintoonenetworkwhileretainingtheinuentialpropertiesoftheoriginalnetworkspartiallyorfully.Aftercouplingthenetworks,wecanexploitexistingsolutionsonthesinglenetworktosolvetheproblem.ThisisapowerfulandcomprehensiveproceduretostudyMIP.Moreover,weproposeanewmetriccalledinuencerelaytoanalyzetheowofinuencebetweennetworks.Throughcomprehensiveexperiments,wediscovercrucialpropertiesofthemultiplenetworksindiffusingtheinformation.Thechapterisorganizedasfollows.InSection 4.1 ,wepresenttheinuencepropagationmodelonmultiplenetworkanddenetheproblem.WethenintroducethemethodtoalignnodesinnetworksinSection 4.2 .Afterthat,weintroducesdifferentcouplingschemesinSection 4.3 and 4.4 .WenextpresenttheinuencerelaytostudytheinuencepropagationprocessinSection 4.5 .Section 4.6 showsexperimentalresults.Inaddition,wepresentcouplingschemesfortwostochasticcascadingmodelsinSection 4.7 .Finally,wesummarizethechapterinSection 4.8 4.1NetworkModelandProblemDenition 4.1.1GraphNotationsWeconsiderknetworksG1,G2,...,Gk,eachofwhichismodeledasaweighteddirectedgraphGi=(Vi,Ei,i,Wi).ThevertexsetVi=fu'sgrepresentstheparticipationofni=jVijusersinthenetworkGi,andtheedgesetEi=f(u,v)'sgcontainsmi=jEijorientedconnections(e.g.,friendshipsorrelationships)amongnetworkusers.Wi=fwi(u,v)'sgisthe(normalized)weightfunctionassociatedtoalledgesintheithnetwork.Inourmodel,weightwi(u,v)canalsointerpretedasthe 67

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strengthofinuence(orthestrengthoftherelationship)auseruhasonanotheruservintheithnetwork.ThesetsofincomingandoutgoingneighborsofvertexuinnetworkGiaredenotedbyNi)]TJ /F8 7.97 Tf -3.61 -7.3 Td[(uandNi+u,respectively.Inaddition,eachuseruisassociatedwithathresholdi(u)indicatingthepersistenceofhisopinions.Thehigheri(u)is,themoreunlikelythatuwillbeinuencedbytheopinionsofhisfriends.Furthermore,theusersthatactivelyparticipateinmultiplenetworksarereferredtoasoverlappingusers.Thoseusersareconsideredasbridgeusersforinformationpropagationacrossnetworks.Finally,wedenotebyG1...kthesystemconsistingofknetworks,andbyUtheexhaustivesetofallusersU=[ki=1Vi. 4.1.2InuencePropagationModelWerstdescribethelinearthresholdmodel(LT-model)[ 24 62 ],apopularmodelforinformationandinuencediffusioninasinglenetwork,andthendiscusshowthisLTmodelcanbeextendedtocopewithmultiplenetworks.IntheoriginalLTmodel,eachnetworkuseruiseitherinanactiveorinactivestate:uisinanactivestateifheoriginallyadoptstheinformation,orthetotalinuencefromhisdirectneighborsexceedshisthreshold(u),i.e,Pv2N(u)w(v,u)(u).Otherwise,uisinaninactivestate.Inabigpicture,givenasystemofknetworks,theinformationispropagatedseparatelyineachnetworkandcanonlybetransferredfromonenetworktoanotherviatheoverlappingusersofthesenetworksTheinformationstartstospreadoutfromsetofseedusersSi.e.allusersinShaveactivestateandtheremainingusersareinactive.Attimet,auserubecomesactiveifthetotalinuencefromitsactiveneighborssurpassesitsthresholdinsomenetworki.e.thereexistisuchthat:Xv2Ni)]TJ /F8 5.978 Tf -2.58 -5.08 Td[(u,v2Awi(v,u)i(u)whereAisthesetofactiveusersaftertime(t)]TJ /F5 11.955 Tf 11.95 0 Td[(1).Aftereachtimestep,newinactiveusersareactivatedandtheycontinuetoactivateotherusers.Theprocesswillcontinueuntiltherearenomoreinactiveuserscanbe 68

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activated.Ifwelimitthepropagationtimetod,thentheprocesswillstopaftert=dtimesteps.ThesetofactiveuserscausedbytheseedsetSaftertimedisdenotedasAd(G1...k,S).Notethatdisalsothenumberofhopsinnetworksuptowhichtheinuencecanbepropagatedfromtheseedset,sodiscalledthenumberofpropagationhops. 4.1.3ProblemDenitionInthischapter,weaddressthefundamentalvulnerabilityproblemonmultiplenetworks:theMassiveInuenceproblem.Theproblemaskstondaseedsetofminimumcardinalitywhichinuencesalargefractionofusers,formallydenedasfollows. Denition4. (MassiveInuenceProblem(MIP))GivenasystemofknetworksG1...kwiththesetofusersU,apositiveintegerd,and0<1,theMIPproblemaskstondaseedsetSUofminimumcardinalitysuchthatthenumberofactiveusersafterdhopsaccordingtoLTmodelisatleastfractionofusersi.e.jAd(G1...k,S)jjUj.Whenk=1,wehavethevariantoftheproblemsonasinglenetworkwhichNP-hardtosolve[ 17 ]butitiseasiertodesignheuristicalgorithmonthesinglenetwork.Innextsections,wepresentdifferentcouplingstrategiestoreducetheproblemonmultiplenetworkstooneonasinglenetworkinordertoutilizethealgorithmdesigning. 4.2NetworkAlignmentWerstreassignauniversalidentication(id)toeachnodeinthenetworkssuchthatalloverlappingnodesofthesameuserhavethesameid.Eachnetworktopologyoftenusesitsownsystemfornamingnodes,thusapersonmayhavedifferentidsindifferentnetworks.Asaconsequence,itrequiresacomplicatedmechanismandextraeffortrepeatedlytokeepupdatingtheuserstatesacrossnetworks.Weeasethisburdenbyassigningauniqueidtoeachuserandusingitasthenodeidinallnetworks.However,ifwetriviallyassignnewidtoanunassignednodeanditsoverlappingnodesonebyone,weneedtoscanalltheoverlapmappingseachtimewhichisalmost 69

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impracticalinlargenetworks.Thus,weneedtodesignanalgorithmwhichassignsnewidsinonlylineartime.Ourgoalistoscaneachoverlapmappingandcheckeachnodeoncetoassignnewids.Insteadofassigningidstoalloverlappingnodesofauseratthesametime,wecheckwhetheroneofitsoverlappingnodesisalreadyassignedanidornot.Tobespecic,weprocesseachnetworkintwophases:assignidstonodesinitsmappinglistswiththeprocessednetworksandthenassignnewidstotheremainingnodes.ThismethodguaranteesthevalidityofnewidsasstatedinLemma 3 .Thealgorithm,asdescribedinAlgorithm 10 ,scanseachmappingandcheckseachnodeonce,sothetotalrunningtimeislinearinthetotalsizeofnetworksandoverlapmappings. Algorithm10Node-AlignmentAlgorithm Require: knetworksG1...kandoverlapmappingsfCijg. Ensure: AnewidmappingfornodesinG1...k. 1: newid 0 2: InitializeidmappingM 3: fori=1tokdo 4: forj=1toi)]TJ /F5 11.955 Tf 11.95 0 Td[(1do 5: foreachpair(u,v)2Cijdo 6: M[u]=M[v] 7: endfor 8: endfor 9: foreachu2Vido 10: ifuisunassignedthen 11: M[u]=newid 12: newid newid+1 13: endif 14: endfor 15: endfor 16: ReturnM Lemma3. Node-AlignmentAlgorithmassignsthesameidtoalloverlappingnodesofthesameuseranddifferentidstonodesofdifferentusers. 70

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Proof. Letui1,ui2,...,uilbeoverlappingnodesofuseruinnetworksi1
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to.Thesedummynodesareisolated.NowthevertexsetViofithnetworkcanberepresentedbyVi=fui1,ui2,...,uingwhereU=fu1,u2,...,ungisthesetofallusers.Inthenewrepresentation,thereisanedgefromuiptouiqifupanduqareconnectedinGi.NowwecanunionallknetworkstoformanewnetworkG.Theapproachtoovercomethesecondchallengeistoallownodesu1,u2,...,ukofanuserutoinuenceeachothere.g.addingedge(ui,uj)withweight(uj).Whenuiisinuenced,ujisalsoinuencedinthenexttimestepastheyareactuallyasingleoverlappinguseru,thustheinformationistransferredfromnetworkGitoGj.Butanemergedproblemisthattheinformationisdelayedwhenitistransferredbetweentwonetworks.Rightafterbeingactivated,uiwillinuenceitsneighborswhileujneedsonemoretimestepbeforeitstartstoinuenceitsneighbors.Itwouldbebetterifbothuiandujstarttoinuencetheirneighborsinthesametime.Forthisreason,newgatewaynodeu0isaddedtoGsuchthatbothuiandujcanonlyinuenceothernodesthroughu0.Inparticular,alledges(ui,vi)((uj,zj))willbereplacedbyedges(u0,vi)((u0,zj)).Inaddition,moreedgesareaddedbetweenu0,ui,andujtolettheminuenceeachother.Wedescribethecouplingschemesnextandhowwecancouplethemultiplenetworkspreservingtheirindividualproperties. 4.3.1CliqueLosslessCouplingSchemeGivenknetworksG1,G2,...,GkwiththesetofusersU,weconstructanewgraphG=(V,E,,w)asfollows.Firstly,weadddummyverticesofthreshold1toallthesenetworksandincludeallnodesintovertexsetVtogetherwithgatewaynodesV=[ki=1Vi[fu01,u02,...,u0ng.Inthenewvertexset,u01,u02,...,u0nrepresentsthesetofusersinthecouplednetworkandarecalleduservertices.VertexuipcalledtheaccountvertexofuserupinGi.Thethresholdsoftheformertypeofverticesaresetto(u0p)=1,1pn,andthethresholdsofthelatertypeverticesarekeptthesamewiththeoneinmultiplenetworksi.e.(uip)=i(up). 72

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Secondly,werepresenttheinuenceofuseruonuservinnetworkGibytheinuenceofuservertexu0ontheaccountvertexofvinGi.ItmeansthatifthereisanedgebetweenuseruandvinGi,thenanedgefromu0toviwithweightw(u0,vi)=wi(u,v)isaddedtotheedgesetE.Finally,weconnectuservertexandaccountverticesofthesameusertoguaranteethattheyhavesameactivationstates.Thegoalisthatifoneofthesenodesisactive,itwillactivateallothernodes.Itcanbedonebyaddingextraedgescalledsynchronizationedgesbetweenthesenodeswhoseweightsequaltothethresholdsofdestinationnodes.Specically,w(ui,uj)=(uj),80i,jk,i6=j.Thesesynchronizationedgesformacliquebetweennodes,thusthiscouplingschemeisnamedcliquelosslesscouplingscheme.AsimpleexampleoftheschemeisillustratedinFigure 4-3 .Nextwewillshowthatthepropagationprocessintheoriginalmultiplenetworksandthecouplednetworkisactuallythesame.Inuenceisalternativelypropagatedbetweenuserandaccountvertices,soproblemwithdhopsinthemultiplenetworksisequivalenttoproblemwith2dhopsinthecouplednetwork. Lemma4. SupposethatthethepropagationprocessonthecouplednetworkGstartsfromtheseedsetwhichcontainsonlyuserverticesS=fs01,...,s0pg,thenuserverticesisonlyactivatedinevenpropagationhops. Proof. Supposethatauservertexu0istherstuservertexthatisactivatedattheoddhops2d+1.u0mustbeactivatedbysomevertexuianduiisthetherstactivatedvertexamongverticesu1,u2,...,uk.Itmeansthatuiisactivatedinhop2d.Sinceallincomingneighborsofuiisuservertices,someuservertexchangesitsstatustoactiveinhop2d)]TJ /F5 11.955 Tf 11.95 0 Td[(1.Itiscontradicted. Lemma5. SupposethatthethepropagationprocessonG1...kandGstartsfromthesameseedsetS,thenfollowingconditionsareequivalent: (1) UseruisactiveafterdpropagationhopsinG1...k. 73

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AMultiplenetworksG1,G2,G3with6userswhereverticeswiththesamecolorrepresentthesameuser.Eachusermayhavedifferentthresholdsindifferentnetworks,e.g.,reduserhasthresholdsof0.6,0.3,and0.8. BTheinuencebetweenusersinmultiplenetworksareen-codedbytheinuencefromuserverticestoaccountvertices.Dummyaccountverticesareaddedtoguaranteethatallusershavethesamenumberofaccountvertices. CCliqueSynchronization Figure4-3. Anexampleoflosslesscouplingscheme 74

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(2) Thereexistsisuchthatuiisactiveafter2d)]TJ /F5 11.955 Tf 11.96 0 Td[(1propagationhopsinG. (3) Vertexu0isactiveafter2dpropagationhopsinG. Proof. Wewillprovethislemmabyinduction.Supposeitiscorrectforany1dt,weneedtoproveitiscorrectford=t+1.DenoteA1...k(t)andA(t)asthesetofactiveusersandactiveverticesaftertpropagationhopsinG1...kandG,respectively.(1))(2):Ifuseruisactiveattimet+1inG1...k,itmustbeactivatedatsomenetworkGj.Wehave:Xv2Nj)]TJ /F8 5.978 Tf -2.76 -5.25 Td[(u\A1...k(t)wj(v,u)j(u)Duetotheinductionassumption,foreachv2A1...k(t),wealsohavev02A(2t)inG.Thus:Xv02N)]TJ /F8 5.978 Tf -0.58 -8.13 Td[(uj\A(2t)w(v0,uj)=Xv2Nj)]TJ /F8 5.978 Tf -2.77 -5.25 Td[(u\A1...k(t)wj(v,u)j(u)=(uj)Itmeansthatujisactiveafter(2(t+1))]TJ /F5 11.955 Tf 11.95 0 Td[(1)propagationhops.(2))(3):Ifthereexistsisuchthatuiisactiveafter2(t+1))]TJ /F5 11.955 Tf 12.12 0 Td[(1propagationhopsonG,thenuiwillactivateu0inhop2(t+1)(3))(1):Supposethatu0=2Sisactiveafter2(t+1)propagationhopsinG,thentheremustexistsujwhichactivatesu0before.Thisisequivalentto:Xv2N)]TJ /F8 5.978 Tf -0.58 -8.13 Td[(uj,v2A(2t)w(v,uj)(uj)Foreachv2A(2t),wealsohavev2A1...k(t).Replacethisintotheaboveinequalitywehave:Xv2Nj)]TJ /F8 5.978 Tf -2.77 -5.26 Td[(u\A1...k(t)wj(v,u)=Xv02N)]TJ /F8 5.978 Tf -0.58 -8.13 Td[(uj\A(2t)w(v0,uj)(uj)=j(u)Thus,uisactiveinnetworkGjaftert+1propagationhops. 75

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Next,wewillshowthatthenumberofinuencedverticesinthecouplednetworksisalways(k+1)timesthenumberofinuencedusersinmultiplenetworksasstatedinTheorem 4.1 Theorem4.1. GivenasystemofknetworksG1...kwiththeusersetU,thecouplednet-workGproducedbythelosslesscouplingscheme,andaseedsetS=fs1,s2,...,spg,ifAd(G1...k,S)=fa1,a2,...,aqgisthesetofactiveuserscausedbySafterdpropagationhopsinmultiplenetworks,thenA2d(G,S)=fa01,a11,...,ak1,...,a0q,a1q,...,akqgisthesetofactiveverticescausedbySafter2dpropagationhopsinthecouplednetwork. Proof. Foreachuserai2Ad(G1...k,S)i.e.aiisactiveafterdhopsinG1...k,thenthereexistsajiwhichisactiveafter2d)]TJ /F5 11.955 Tf 12.8 0 Td[(1hopsinGaccordingtotheLemma 5 .Asaconsequence,alla0i,a1i,...,akiareactiveafter2dhops.SoB=fa01,a11,...,ak1,...,a0q,a1q,...,akqgA2d(G,S).LetconsideravertexofA2d(G,S)whichis:Case1.Auservertexu0whichisactiveafter2dhopsinG,sovertexumustbeactiveafterdhopsinG1...k.Thisimpliesu2Ad(G1...k,S),thusu02B.Case2.Anaccountvertexui.Ifuiisactiveafter2d)]TJ /F5 11.955 Tf 12.03 0 Td[(1hops,thenumustbeactiveafterdhopsduetoLemma 5 ,thusu2Ad(G1...k,S).Otherwise,uiisactivatedathop2d,itmustbeactivebysomevertexuj,j>0sincealluserverticesonlychangetheirstateatevenhops.Again,u2Ad(G1...k,S).Thisresultsinui2B.Fromtwoabovecases,wealsohaveA2d(G,S)B.SothatA2d(G,S)=B,theproofiscompleted. Theorem 4.1 providesthebasistoderivethesolutionforMIPonmultiplenetworksfromthesolutiononasinglenetwork.ItimpliesanimportantalgorithmicpropertyofthelosslesscouplingschemeregardingtotherelationshipbetweenthesolutionsofMIPinG1...kandG.Theequivalenceoftwosolutionsisstatedbelow: 76

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Theorem4.2. Whenthelosslessschemeisused,thesetS=fs1,s2,...,spginuencesfractionofusersinG1...kafterdpropagationhopsifandonlyifS0=fs01,s02,...,s0pginuencesfractionofverticesincouplednetworkGafter2dpropagationhops.Sizeofthecouplednetwork.Thesizeofthecouplednetworkcanbecomputedfromthesizesoftheoriginalnetworksasfollows: Proposition4.1. Whenthelosslessschemeisused,thecouplednetworkhasjVj=(k+1)jUj=(k+1)nverticesandjEj=Pki=1jEij+nk(k+1)edges. Proof. Inthecouplingscheme,eachuseruhask+1correspondingverticesu,u1,...,ukinthecouplednetwork,thusthenumberofverticesisjVj=(k+1)jUj=(k+1)n.Thenumberofedgesequalsthetotalnumberofedgesfromallinputnetworksplusthenumberofnewedgesforsynchronizing.ThusthetotalnumberofedgesisjEj=Pki=1jEij+nk(k+1). 4.3.2StarLosslessCouplingSchemeIncliquelosslesscouplingscheme,thenumberofedgestosynchronizethestateofverticesu0,u1,...,ukisk(k+1)foreachuseru,whichresultsinnk(k+1)extraedgesinthecouplednetwork.Inrealnetworks,thenumberofedgesisoftenlineartothenumberofvertices,sothenumberofextraedgesconsiderablyincreasesthesizeofthecouplednetwork,especiallywhenkislarge.Wewouldliketodesignasynchronizationstrategythatreducestheseextraedges.Notethatthelargenumberofextraedgesisduetothedirectsynchronizationbetweeneverypairsofaccountverticesofuincliquelosslesscouplingscheme,sowecansavesomeedgesbyusingindirectsynchronization.Wecreateonemoreintermediatevertexuk+1withthreshold(uk+1)=1andlettheactivestatepropagatefromanyvertexinu1,u2,...,ukviathisvertex.Specically,thesynchronizationedgesareestablishedfollows:w(ui,uk+1)=1andw(uk+1,ui)=(ui)1ik;w(uk+1,u0)=w(u0,uk+1)=1.Thesynchronizationstrategyofstarlosslesscouplingschemeis 77

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Figure4-4. StarSynchronization illustratedinFig. 4-4 .Now,thenumberofextraedgeforeachuseris2(k+1)andthesizeofthecouplednetworkisreducedto: Proposition4.2. Whenstarlosslessschemeisused,thecouplednetworkhasjVj=(k+2)jUj=(k+2)nverticesandjEj=Pki=1jEij+2n(k+1)edges.Instarlosslesscouplingscheme,ittakes2hopstosynchronizethestatesofaccountverticesofeachuserwhichleadstodelayingthepropagationofinuenceinthecouplednetwork.Duetothesimilaritybetweenstarlosslessschemeandcliquelosslessscheme,westatethefollowingpropertyofstarlosslessschemewithoutproof. Theorem4.3. Whenstarlosslesscouplingschemeisused,thesetS=fs1,s2,...,spginuencesfractionofusersinG1...kafterdpropagationhopsifandonlyifS0=fs01,s02,...,s0pginuencesfractionofverticesincouplednetworkGafter3dpropagationhops. 4.4LossyCouplingSchemesIntheprecedingcouplingschemes,acomplicatedcouplednetworkisproducedwithalargenumberofauxiliaryverticesandedges.Itisidealtohaveacouplednetworkwhichonlycontainusersasnodes.Thisnetworkprovidesacompactviewoftherelationshipbetweenuserscrossingthewholesystemofnetworks.Tocompactthe 78

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informationwhichiscompletelydescribedbythewholesystemintoonenetwork,thelossofinformationisunavoidable.Thegoalistodesignaschemesuchthatminimizethelossasmuchaspossiblei.e.thesolutionfortheprobleminthecouplednetworkisveryclosedtooneintheoriginalsystem.Next,wepresentsuchschemebasedonthefollowingkeyobservations.Observation1.Consideruseru,uwillbeactivatedifthereexistsisuchthat:Xvi2N)]TJ /F8 5.978 Tf -0.57 -8.14 Td[(ui,v2Awi(vi,ui)i(u)whereAisthesetofactiveusers.Wecanrelaxtheconditiontoactivateuwithpositiveparameters1(u),2(u),...,k(u)asfollows: kXi=1(i(u)Xvi2N)]TJ /F8 5.978 Tf -0.58 -8.13 Td[(ui,v2Awi(v,u))kXi=1i(u)i(ui)(4) Proposition4.3. GivenasystemofnetworksG1...k,ifthecondition( 4 )issatised,thenuseruisactivated. Proof. Whentheconditionissatised,theremustexistisuchthati(u)Pvi2N)]TJ /F8 5.978 Tf -0.58 -8.13 Td[(ui,v2Awi(v,u)i(u)i(u).Asaconsequence,theconditiontoactivateuissatisedsincei(u)>0 Notethatsometimestheconditiontoactivateuismet,butthecondition( 4 )isstillneedmoreinuencefromu'sfriendstosatisfy.Themorethisextrainuenceneedis,theloosercondition( 4 )is.Wecanreducethisredundancybyincreasingthevalueofi(u)proportionaltothevalueofPvi2N)]TJ /F8 5.978 Tf -0.58 -8.13 Td[(ui,v2Awi(v,u))]TJ /F12 11.955 Tf 12.72 0 Td[(i(u).Inthespecialcase,ifPvi2N)]TJ /F8 5.978 Tf -0.58 -8.13 Td[(ui,v2Awi(v,u)>i(u)andwechoosei(u)j(u),8j6=i,thenthereisnoredundancy.Unfortunately,wedonotknowbeforehandinwhichnetworkuseruwillbeactivated,sowecanonlychooseparametersheuristically.Observation2.Whenuseruparticipatesinmultiplenetworks,itiseasiertoinuenceuinsomenetworkthantheothers.Thefollowingsimplecaseillustratesuchsituation.Supposethatwehavetwonetworks.Innetwork1,1(u1)=0.1and 79

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Figure4-5. Lossycouplednetworkusingeasinessparameters.Thenumberofedgesismuchlessthanthelosslesscouplednetwork. u1has8in-neighbors,eachneighborv1inuencesu1withw1(v1,u1)=0.1.Innetwork2,2(u2)=0.7andu2has8in-neighbors,eachneighborv2inuencesu2withw2(v2,u2)=0.1.Thenumberofactiveneighborstoactivateuis1and7innetwork1and2,respectively.Intuitively,wecansaythatuiseasiertobeinuencedintherstnetwork.Wequantifytheinuenceeasinessi(u)thatuisinuencedinnetworkiastheratiobetweenthetotalinuencefromfriendsandthethresholdtobeinuenced.i(u)=Pvi2N)]TJ /F9 7.97 Tf 6.25 -2.27 Td[((ui)wi(vi,ui) i(ui)Wecanusetheinuenceeasinessofauserinnetworksastheparametersofthecondition 4 .Basedonaboveobservations,wecouplemultiplenetworksintooneusingparametersfi(u)g.ThevertexsetisthesetofusersV=fu1,u2,...,ung.Thethresholdofvertexuissetto:(u)=kXi=1i(u)i(ui) 80

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Theweightoftheedge(v,u)is:w(v,u)=kXi=1i(u)wi(vi,ui)wherewi(vi,ui)=0ifthereisnoedgefromvitouiinithnetwork.ThenthesetofedgesisE=f(v,u)jw(v,u)>0g.Fig. 4-5 illustratestheloosycouplednetworkofthesystemofnetworkinFig. 4-3 .Besidestheeasiness,othermetricscanbeusedwiththesamepurpose.Weenumerateheresomeothermetrics.Involvement.Nodescanbepartofmultiplesocialnetworks,buttypicallytheyaremoreinvolvedinafewofthemcomparedtoothers.WeestimateinvolvementofanodevinanetworkGibymeasuringhowstronglythe1-hopneighborhoodvisconnectedandtowhatextentinuencecanpropagatefromonenodetoanotherinthe1-hopneighborhood.FormallywecandeneinvolvementofanodevinnetworkGias:-iv=Xx,y2fNi(v)[vgwi(x,y) iywhereNi(v)isthesetofallneighborsofv(bothin-comingandout-going),wi(x,y)isthewtofedge(x,y)andiyisthethresholdofyinGi.Average.Thisabaselineschemejustusedforcomparisonpurposes.Wejusttakeanaverageofthethresholdsandedge-wtsoverallthenetworks,inwhichvbelongs.Soaverageofanodevinnetworkicanbedenedas iv=1 jP(v)jNextweshowtherelationshipbetweenthesolutionfortheinuencemaximizationprobleminthelossycouplednetworkandtheoriginalsystemofnetworks.Asdiscussedintheaboveobservations,ifthepropagationprocessstartsfromthesamesetofusersinthenetworksystemG1...kandthecouplednetwork,thentheactivestateofauserinGimpliesitsactivestateinG1...k.ItmeansthatifthesetofusersSactivatesfraction 81

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ofusersinG,italsoactivatesatleastfractionofusersinG1...k.Wehavethefollowingresult. Theorem4.4. Whenthelossycouplingschemeisused,ifthesetofusersSactivatesfractionofusersinG,thenitactivatesatleastfractionofusersinG1...k. 4.5InuenceRelayWeproposetheinuencerelaymetrictoquantifytheroleofusersinpropagatinginformation.Whentheinformationisdiffusedinmultiplenetworks,theinformationmayowwithinasinglenetworkorgothroughtwoormorenetworks.Thisbringsoutaseriesofconcerns:howmuchinformationowsinsideanetwork?howmuchinformationowsfromonenetworktoanother?howmuchisthecontributionofeachnetworkintheinuencepropagation?Oncewecanquantifythesevalues,wecangetinsightsintotheinuencediffusionprocessinmultiplenetworks.Next,wedenetheinuencerelaymetricandrelatedconceptstomeasurethesevalues.Sincewecanuseasinglenetworktosimulatethediffusionprocessinmultiplenetworks,werstcanmeasuretheinformationowingthrougheachnodeinasinglenetwork.SupposethattheinformationbreaksoutfromtheseedsetSinthenetworkG,andstopsafterdhopswiththesetofinuencedverticesAd(G,S).Intuitively,theinuencerelayofeachvertexistheamountofinuenceitrelaystoothernodesafteradoptingtheinformation.Themorenumberofverticesithelpstoinuence,thehigheritsinuencerelayis.Inaddition,ifithasstronginuenceonanodewithhighrelayinuence,itshouldalsohavehighvalueofrelayinuenceevenitdoesnotdirectlyinuencemanyvertices.Forthesereasons,weformallyproposetheinuencerelaymetricIR()whichiscomputediterativelyasbelow.Allinactiveverticeshavetheinuencerelayof0.Eachactivevertexvwithoutactivatedoutgoingneighborshastheinuencerelayof1.vdoesnotactivateanyvertices,butitcontributesitselfasoneactivenodetoAd(G,S). 82

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Theinuencerelayofanyothervertexuiscomputedbasedontheinuencerelayofitsoutgoingactiveneighbors.Specically,theinuencerelayofuis: IR(u)=1+Xv2N+u^h(u)h(u).ThegraphIGSisadirectedacyclicgraph,thuswecancomputetheinuencerelayofallverticesinthereversetopologicalorderingofIGSasdescribedinCIRalgorithm(Algorithm 11 ). Proposition4.4. TheinuencegraphIGScausedbytheseedsetSinthenetworkGisadirectedacyclicgraph. Proof. IfIGShasacycleu1,u2,...,ut,u1,thenh(u1)
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Algorithm11ComputingInuenceRelay(CIR) Require: AnetworkG,aseedsetSandthenumberofhopsd. Ensure: TheinuencerelayIRofallvertices. IGS TheinuencegraphcausedbySonG foreachu2VSdo IR(u) 0 endfor Computethetopologicalorderingu1,u2,...,unSofverticesinVS fori=nSdownto1do IR(ui) IR(ui)+1 total 0 foreachv2N)]TJ /F5 11.955 Tf 7.08 -4.34 Td[((ui)do total total+w(v,ui) endfor foreachv2N)]TJ /F5 11.955 Tf 7.08 -4.34 Td[((ui)do IR(v) IR(v)+w(v,ui)IR(ui) total endfor endfor ReturnIR Lemma6. TheCIRalgorithmproducestheinuencerelayforeachactivatedvertex. Proof. Weuseinductiontoprovethattheinuenceukiscomputedcorrectlyaftertheloopi=k.Firstly,uniscomputedrstandIR(un)=1.Thisiscorrectsinceunisattheendofthetopologicalorderinganddoesnothaveanyactivatedoutgoingneighbors.Now,supposethattheinuencerelayofun,un)]TJ /F9 7.97 Tf 6.58 0 Td[(1,...,uk+1iscomputedcorrectlyaftertheloopi=k+1,wewillprovethatIR(uk)holdstheinuencerelayofukaftertheloopi=k.Letfui1,ui2,...,uipgbethesetofactivatedoutgoingneighborsofukwhichisactivatedlaterthanuk.DuetotheconstructionofIGS,(uk,ui1),(uk,ui2),...,(uk,uip)areedgesinIGS,henceiq>k,1qp,inthetopologicalordering.Afterthei=iqloop,IR(uk)willreceiveavalueofw(uk,uiq)IR(uiq) Pz2N)]TJ /F8 5.978 Tf -0.43 -4.82 Td[(uiq^h(z)
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Timecomplexity.ThetopologicalorderingofadirectedacyclicgraphcanbecomputedinlineartimeandthenumberofupdatesinthemainloopequalstothenumberofedgesofIGS,sotheCIRalgorithmrunsinlineartime.AcrucialpropertyofthenewmetricisthatthetotalinuencerelayofseedverticesreectstheinuenceoftheseedsetasstatedinTheorem 4.5 Theorem4.5. Thetotalinuencerelayofseedingverticesequalsthetotalnumberofactivatedvertices.Xu2SIR(u)=jAd(G,S)j Proof. TheproofisbasedonaninvariantofvariablesIR(u1),...,IR(un)inCIRalgorithm.Theinformationispropagatedfromtheseedset,thusallseedverticesdonothaveincomingneighborsinIGSandoccupysmallestindicesinthetopologicalordering.Letupbethehighestindexseedvertex.Wewillprovethataftertheloopi=k+1wehave:kXj=1IR(uj)=nS)]TJ /F3 11.955 Tf 11.95 0 Td[(k,8pknSBeforetheloopi=n,itisobviouslytrue.Aftertheloopi=k,thevalueofvariableIR(uk+1)isincreasedby1andredistributedtoitsincomingneighbors,thusPk)]TJ /F9 7.97 Tf 6.58 0 Td[(1j=1IR(uj)equalsPkj=1IR(uj)plus1.ItimpliesthatPk)]TJ /F9 7.97 Tf 6.58 0 Td[(1j=1IR(uj)=nS)]TJ /F5 11.955 Tf 11.96 0 Td[((k)]TJ /F5 11.955 Tf 11.96 0 Td[(1)aftertheloopi=k.Aftertheloopi=p+1,wehavePpi=1IR(ui)=nS)]TJ /F3 11.955 Tf 12.16 0 Td[(p.Ateachloopi=pdowntoi=1,thevalueofIR(ui)isincreasedby1.Thus,whenthealgorithmstopswehave:Xu2SIR(u)=pXi=1IR(ui)=nS)]TJ /F3 11.955 Tf 11.96 0 Td[(p+p=jAd(G,S)j Theorem 4.5 impliesthateachvertexu2ScontributesIR(u)inpropagatingtheinuenceoverthenetworkG.Now,supposethatGisthecouplednetworkofmultipleones,wecansumuptheinuencerelayofallseedverticesofacomponentnetworktoobtainthecontributionofthatnetwork.Furthermore,thetotalinuencerelay 85

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ofoverlappingverticesindicatestheamountinformationpropagatedbackandforthbetweennetworks.Wecanalsoadapttheinuencerelaymetrictomeasurethesupportbetweennetworksinpropagatinginformation.Let'sconsiderthecasetheinformationemergesfromtheseedverticesofonenetwork,propagatestoanothernetworksviaoverlappingusers,thencomesbacktotherstnetwork.Withthesupportofothernetworks,theinformationispropagatedfurtherintherstnetwork.Ifweconsiderthediffusionprocessinthecouplednetworkandincreasetheinuencerelayby1onlyonactivatedverticesintherstnetwork,wecanquantifythesupportfromothernetworksbythetotalinuencerelaywhichgoesthroughothernetworks. 4.6ExperimentalEvaluationInthissection,weshowtheexperimentalresultsforcouplingschemesandusethecouplingschemestoanalyzetheinuencediffusioninmultiplenetworks.Firstly,wecomparelosslessandlossycouplingschemestomeasurethetrade-offbetweentherunningtimeandthequalityofsolutions.SincethemassiveinuenceproblemisNP-hard[ 17 ]inasinglenetwork,weusethegreedyalgorithm,whichprovideshighqualitysolution,tondthesolutionaftercouplingnetworks.Wealsoinvestigatetherelationshipbetweennetworksintheinformationdiffusiontoanswerthefollowingquestions:(1)Whatistheroleofoverlappingusersinthediffusionoftheinformation?(2)Howdoesanetworkgetbenetfromothernetworkstodiffusetheinformation?(3)Canthediffusionononenetworkprovideaburstofinformationinothernetworks?(4)Whatwillwemissifweconsidereachnetworkseparately?WeranallourexperimentsonIntel(R)Xeon(R)CPUW350machinewitha12GBRAManda2.93GHzQuad-coreprocessor.Inallexperiments,bydefault,thenumberofhopsisd=4andtheinuencefraction=0.8. 86

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Networks#Nodes#EdgesAvg.Degree Twitter4827716304712289.7 FSQ44992166440235.99 CM404201756928.69 Het8360157511.88 NetS158827421.73 Table4-1. Foursquare-Twitterandco-authornetworkdata-sets 4.6.1DatasetsRealnetworks.Wedoexperimentsontwosystemsofnetworks:TwitterandFoursquare(FSQ)networks[ 49 ],andco-authornetworksintheareaofCondensedMatter(CM)[ 43 ],High-EnergyTheory(Het)[ 43 ],andNetworkScience(NetS)[ 42 ].ThestatisticsofnetworksaredescribedinTable 4-1 .Whiletheoverlappingusersoftherstdatasetisprovidedin[ 49 ],wematchoverlappingusersofthesecondonebasedonauthors'names.ThenumbersofoverlappingnodesofnetworkpairsFSQ-Twitter,CM-Het,CM-NetS,andHet-NetSare4100,2860,517,and90,respectively.Moreover,whileco-authornetworkshaveedgeweights,FSQ-Twitterdatasetonlycontainsthenetworktopologies.Ifthenetworkdoesnothaveedgeweights,weassigntheweightofeachedgerandomlyfrom0to1.Wethennormalizetheedgeweightssuchthatthetotalweightofin-comingedgesis1foreachnode.Thisissuitablesincetheinuenceofuseruonuservtendstobesmallifvisundertheinuencemanyfriends.Finally,thethresholdofeachnodeisarandomvaluefrom0to1.Synthesizednetworks.WealsousesynthesizednetworksgeneratedbyErdos-Renyirandomnetworkmodel[ 26 ]totestnetworkswithcontrolledparameters.Therearetwonetworkswith5000nodesareformedbyrandomlyconnectingeachpairofnodeswithprobabilityp1=0.0008andp2=0.006.Theaveragedegrees,8and60,reectthediversityofnetworkdensitiesinthereality.Then,weselectrandomlyffractionofnodes 87

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intherstnetworkasoverlappingnodes.Theedgeweightsandnodethresholdsareassignedasabove. 4.6.2ComparisonofCouplingSchemesWerstevaluatetheeffectofthecouplingschemesontherunningtimeandthequalityofthefoundsolutionswhenweusethegreedyalgorithmtosolveMIP.AsillustratedinFig. 4-6 ,thealgorithmprovideslargerseedsetsbutrunsfasterinlossycouplednetworksthanlosslesscouplednetworks.InbothTwitter-FSQandco-authordatasets,theseedsizesaresmallestwhenthelosslesscouplingschemeisused.ItisasexpectedsincethelosslesscouplingschemereservesalltheinuenceinformationwhichisexploitedlatertosolveMIP.However,theseedsizesareonlyabitlargerusingthelossycouplingschemes.Inthelossycouplingschemes,theinformationisonlylostatoverlappinguserswhichoccupiesasmallfractionthetotalnumberofusers(roughly5%inTwitter-FSQand7%inco-authornetworks).Thus,theeffectoflossycouplingschemesonthesolutionqualityissmallespeciallywhentheseedsetsarebigtoinuencealargefractionofusers.Ontheotherhand,thealgorithmrunsmuchfasterinlossycouplednetworkswiththefactorupto2timesinTwitter-FSQand4timesinco-authornetworks.Themajordisadvantagesofthelosslesscouplingschemeisthedoublednumberofhops,thenumberofextranodesandedges.Inco-authordataset,thenumberofextraedgesarerelativehighcomparingtothetotalnumberofedgesinallnetworks,sothespeedingupfactorishigherinco-authornetworks.Wethereforecaninferthatthelossycouplingschemesworkwellonrealdatasetsinwhichnetworksaresparseandthenumberofoverlappingusersissmall.Next,weexaminetheeffectofthenumberofoverlappingusersontheperformanceofthecouplingschemeswiththesynthesizeddatasets.Fig. 4-7 demonstratestheresultsontwonetworksofsize5000anddifferentfractionofoverlappingusersf.Theoverlappingfractionsignicantlydifferentiatethecouplingschemesintermsofboththesolutionqualityandrunningtime.Whenfissmall,theseedsizesarequiteclosewithallcouplingschemes.But 88

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Figure4-6. ComparingCouplingSchemesforFindingMinimumSeedSetonco-authorNetworks(uppergures)andonFSQandTwitter(lowergures) whenfincreases,thegapbetweenschemesisbiggerandbigger.Thevariationoffalsorevealstheeffectivenessoftheeasinesslossycouplingscheme(thebest)andthedisgraceofthetrivialaveragescheme(theworst)amongthelossyones.Itismoreinterestingwhenwelookattherunningtime.Therunningtimeinthelosslesscouplednetworksisinitiallyhigherthaninthelossycouplednetworksbutitgraduallycatchesupandovertakesthelaternetworksatf=0.4.Thekeypointisthesizeoftheseedset.Thelargerfis,thelargertheratiobetweentheseedsizeinlosslessandlossycouplednetworksis.Astherunningtimedependsontheseedsize,therunningtimeinthelosslesscouplednetworkreducesfaster.Thus,werecommendtouselosslessschemewhentheoverlappingfractionislargeandtheseedsizeispredictedtobesmall. 89

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Figure4-7. Comparingcouplingschemeswithdifferentoverlappingfractionf ACo-authornetworks BFSQandTwitter Figure4-8. Comparingcouplingschemeswithdifferentnumberofpropagationhopsd Sincetheseedsizeissensitivetothenumberofhops[ 24 ],wewouldliketoevaluatecouplingschemeswithdifferentpropagationhops.Similartosinglenetworks,Fig. 4-8 showsthattheseedsizedecreaseswhenwehavelargernumberofpropagationhops.However,thelossycouplingschemesdeviatemoreandmorefromthelosslessoneintermsoftherelativeseedsizewhenthenumberofhopsincreases.Let'sconsidertheratiooftheseedsizesbetweenthebestlossycouplingscheme(theeasinessone)andthelosslesscouplingscheme.Itis1.05(1.1)and1.5(1.3)inco-authornetworks(FSQ-Twitter)withd=2andd=5.Thereasonisthatthelossycouplingschemesinherentlybeartheerrorwhichisaccumulatedandpropagatedaftereachhop. 90

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4.6.3BenetsofCoupledNetworkCouplingschemesprovidethemechanismtostudymultiplenetworksunderaconsistentviewwhichhelpstoanswerdifferentconcernsabouttheinuencediffusion.Forexample,Fig. 4-6 showsapropertythatissimilartooneinasinglenetwork:theseedsizeincreasessuperlinearlyregardingtotheinuencefraction.Itmeansthatthegainperseedusersisdecreasedwhenthecircleofinuenceisbroadened.Moreover,withoutthecouplednetwork,wemayneedtondtheseedsetoneachnetworktoinuencefractionofallusersandunionthemtoobtaintheseedsetforthewholesystem.Fig. 4-9 clearlydemonstratesthatifweinuenceeachnetworkseparatelywewouldneedamuchlargerseedsetcomparedtowhatweneedinthecouplednetwork,nomatterwhichtypeofcouplingweuse.Theseedsetfoundonthelosslesscouplednetworkalmosthasthesamesizewiththelargestseedsetfoundincomponentnetworksinco-authordatasetandevensmallerinTwitter-FSQ.Inco-authordatasets,thesizeoftheunionsettoinuence0.8fractionofusersis24%and30%largerthesizeoftheseedsetsfoundinlosslessandlossynetworks.Thesenumbersare23%and47%inTwitter-FSQ.Thereasonisthatthelossless(lossy)couplednetworkcancapture(partiallycapture)thecollaborationofnetworkstopropagatetheinformationandexploitittoreducetheseedsize.Whenwendtheseedsetineachnetworkseparately,weignorethisproperty.Asaconsequence,weendureapenaltyonthesizeoftheunionsetwhichishighifnetworkscanpropagatetheinformationwelllikeTwitterandFSQ.AlthoughwecanuseothermethodstosolveMIPwithoutusingcouplingschemes,theymaybemorecomplicatedandcausetheseedsizeincrease.ThecouplingschemesnotonlyhelptosolveMIP,itisalsohelptoinvestigateotheraspectsoftheinuencediffusioninthesystemofnetworks.originDuetotheoverlappingwithothernetworks,wemayunderestimatetheabilitytodiffusetheinformationofaspecicnetwork.Itmotivatesusgaugetheviralmarketingpotentialofanetworkallowingtheinformationtobepropagatetobackandforthtoothernetworks. 91

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ACo-authornetworks BFSQandTwitter Figure4-9. Thequalityofseedsetswithandwithoutusingthecouplednetwork Specically,weusethegreedyalgorithmtondthesmallestseedsettoinuencefractionofthestudiednetwork'usersinthelosslesscouplednetwork.Thenwecompareittotheseedsetfoundinthetraditionalperspectiveconsideringthisnetworkasastandaloneone.AsshowninFig. 4-10 ,theseedsizedecreasesupto9%,25%,17%,and26%inCM,Het,FSQ,andTwitter,respectively,whenweconsiderthesenetworksintheconnectionwithothernetworks.TheimprovementinNetSissmallduetothesmallnumberofoverlappinguserswithothernetworks.ItisalsoobservedthattheimprovementratioishigherfornetworkwithlowconductanceofinuenceinthecaseofFSQandTwitter,twonetworkswiththesamenumberofusers.Whenthenetworksizesareunbalanced,HetthenetworkwiththesmallernumberofusersseemstogetbetterimprovementratiothanthebiggernetworkCM.Thebackandforthpropagationoftheinformationbetweennetworksisthebasefortheoutsidesupportofthetargetnetwork.Whentheinformationispropagatedfromseednodesinthetargetnetwork,somenodesareactivatedinothernetworksduetotheoverlappingnodes.Theinformationthenispropagatedfurtherandevencomesbacktothetargetnetwork,hencethenumberofinuencednodeinthetargetnetworkisincreased.Fig. 4-11 showstheamountofinuencerelaythatothernetworkssupportthetargetnetworkwithd=4andd=8hops.Thesupportisconsiderableandhigherwiththelargernumberofhops.When 92

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ACo-authornetworks BFSQandTwitter Figure4-10. Thequalityofseedsetswithandwithoutusingthecouplednetwork d=8,theinformationhasmorechancetocomebacktothetargetnetwork;thesupportisupto2.2%,5%,8.3%,and7.3%onthenetworkCM,Het,Twitter,andFSQ.Thesupportishigheriftheinformationiseasiertobepropagatedincomponentnetworks. 4.6.4BiasinSelectingSeedNodesHere,weanalyzethetheseedsetonthelosslesscouplednetworktoobservehowmucheachnetworkcontributestowardsthecompositionoftheseedsetandthesetofinuencednodes.Wemainlyaddresstwoquestions:(1)whichnetworksupportsthepropagationbetterand(2)whetherthereisabiastowardanetworkselectingseednodes.Fig. 4-12 showsthefractionofselectednodesaswellasinuencednodesineachnetworkandtheoverlappingpart.Wecanobservethatoverlappingnodestendtobeselectedinbothdatasets.Whentheinuencedfractionis0.4,thefractionofoverlappingseednodesisaround24.9%and25%onco-authorandFSQ-Twitternetworks,respectively.Notethatonly5%(7%)totalusersofFSQ-Twitter(co-authornetworks)areoverlappingusers.Thisshowsthatoverlappingusersnotonlyplaytheroleasbridgesforinformationtopropagatebetweennetworksbutalsohavehighinuence.AsillustratedinFig. 4-13 ,thecontributionoftheoverlappingnodesininuencingothernodesishigh,especiallywhenissmall.Additionally,thereisanunbalancebetweenthenumberofselectedseedsandinuencednodesineach 93

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ACo-authornetworks BFSQandTwitter Figure4-11. Thesupportbetweennetworksontheinuencepropagationofanetworkwithd=4(uppergures)andd=8(lowergures)hops.C,H,N,F,andTaretheabbreviationsofCM,Het,NetS,FSQ,andTwitter. networks.Inco-authordataset,thebiggestnetwork,CM,contributesalargenumberofseednodesandinuencednodes.When=0.8,76.7%ofseednodesand80.5%ofinuencednodesarefromCM.Incontrast,thenumberofseednodesfromFSQissmallbutthenumberofinuencednodesinFSQmuchhigherthanTwitter.Letconsidertheinuencefractionof0.4,27%(withoutoverlappingnodes)ofseednodesbelongtoFSQwhile70%ofinuencednodesareinFSQ.AfternodesinFSQarealmostinuenced,thealgorithmstartstoselectmorenodesinTwittertoincreasetheinuencefraction.Itimpliesanimportantcharacteristicofmultiplenetworks.Iftheinformationiseasiertoowinonenetwork,thatnetworkwillattractandpropagatemoreinformationinside.In 94

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Figure4-12. Thebiasinselectingseednodesonsynthesizednetworks(uppergures)andonFSQandTwitter(lowergures) thebigpicture,itprovideshintsforviralmarking:overlappingnodeshavehighpotentialtotargetandsomenetworksaremoreefcienttoadvertisethanothers. 4.7ExtensionstoOtherCascadingModelsInthissection,weshowthatwecandesignlosslesscouplingschemesforsomeotherwell-knowncascadingmodelsineachcomponentnetwork.Asaconsequence,topinuentialnodescanbeidentifyunderthesemodels.Inparticular,weinvestigatetwomostpopularstochasticdiffusionmodelswhichareStochasticThresholdmodelandIndependentCascadingmodel[ 34 ]. StochasticThresholdmodel.ThismodelissimilartotheLinearThresholdmodelbutthethresholdi(ui)ofeachnodeuiofGiisarandomvalueintherange[0,i(ui)].NodeuiwillbeinuencedwhenPvi2N)]TJ /F8 5.978 Tf -0.58 -8.13 Td[(ui,v2Awi(vi,ui)i(ui) 95

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ACo-authornetworks BFSQandTwitter Figure4-13. Theinuencecontributionofseednodesfromcomponentnetworks IndependentCascadingmodel.Inthismodel,thereareonlyedgeweightsrepresentingtheinuencebetweenusers.OncenodeuiofGiisinuenced,ithasasinglechancetoinuenceitsneighborvi2N+(ui)withprobabilitywi(ui,vi).Forbothmodels,weusethesameapproachofusinguservertices,accountverticesandthesynchronizationbetweenuserverticesandtheiraccountvertices.Specically,theweightofedge(ui,uj),0i6=jkwillbe(uj)forStochasticThresholdmodeland1forIndependentCascadingmodel.Withthisassignment,ifuiisinuenced,ujwillbeinuencedwithprobability1inthenexttimestep.TheprooffortheequivalenceofthecouplingschemeissimilartoonesforSection 4.3 4.8SummaryInthischapter,westudythemassiveinuenceprobleminmultiplenetworks.Totackletheproblem,weintroducednovelcouplingschemestoreducetheproblemtoaversiononasinglenetwork.Thenwedesignanewmetrictoquantifytheowofinuenceinsideandbetweennetworksbasedonthecouplednetwork.Exhaustiveexperimentsprovidenewinsightstotheinformationdiffusioninmultiplenetworks.Inthefuture,weplantoinvestigatetheproblemonmultiplenetworkswithheterogeneousdiffusionmodels.Inparticular,eachnetworkmayhaveitsowndiffusionmodel,thequestionishowtorepresentthemefciently.Doesthereexistamethodtocouplethemintoonenetwork? 96

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CHAPTER5CONCLUSIONSInthisthesis,westudytheproblemofidentifyinggranularnodesofthecascadingpropagationinnetworks.Underthecascadingeffect,thesenodeshaveastrongimpactoverthenetwork.Itiscrucialtodetectsuchnodestoservevariouspurposese.g.theeconomicalgain.Whenthestudiednetworksaresocialnetworks,wecanusenodesasthetargetforadvertising.Ontheotherhand,ifnetworksareinfrastructureonlikepowernetwork,communicationnetworks,etc.,wecanprotectthesenodesfrombeingattacked.Foreachkindofnetworks,weproposeefcientstrategiestondsuchgroupofnodes.Ininterdependentinfrastructurenetwork,weintroduceanewcentralityforcouplednetworksandutilizeittodetectmostvulnerablenodes.Inaddition,aefcientgreedyframeworkisproposedwherethepuregreedyandcentralitymeasurearecombinedtoprovideabettersolutioninshortertime.Inmultipleonlinesocialnetworks,wedesignanovelframeworktondtopinuentialusers.Inparticular,novelcouplingschemesaredesignedtoreducetheproblemonmultiplenetworkstooneonanetwork.Asaconsequence,wecanapplyexistingsolutionsforanetworktondmostinuentialnodesinmultiplenetworks.Itisacrucialconnectionwhichshowsthatsolvingtheproblemoncouplednetworksisaseasyasonthesinglenetwork.Webelievethatthecouplingschemescanbeextendedtoothermodels.Finally,weinvestigatethecascadingfailureunderloadredistributionmodelinpowernetworks.Anewcascadingcentralityisdesignedspecicallyloadredistributionmodelandcanbeusedtodetectmostcriticalnodesefciently.Moreover,weproposethecooperatingattackstrategytoevaluatetheweaknessofnetworksevenwhenitisdesignedtotoleratenodefailures. 97

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[28] Goldenberg,J.,Libai,B.,andMuller,E.Talkofthenetwork:Acomplexsystemslookattheunderlyingprocessofword-of-mouth.Marketingletters12(2001).3:211. [29] Grubesic,TonyH.,Matisziw,TimothyC.,Murray,AlanT.,andSnediker,Diane.ComparativeApproachesforAssessingNetworkVulnerability.InternationalRegionalScienceReview31(2008).1:88. [30] Hines,Paul,Blumsack,Seth,CotillaSanchez,E,andBarrows,Clayton.Thetopologicalandelectricalstructureofpowergrids.SystemSciences(HICSS),201043rdHawaiiInternationalConferenceon.IEEE,2010,1. [31] Hopcroft,JohnandTarjan,Robert.Algorithm447:efcientalgorithmsforgraphmanipulation.Commun.ACM16(1973).6:372. [32] Huang,Xuqing,Gao,Jianxi,Buldyrev,SergeyV.,Havlin,Shlomo,andStanley,H.Eugene.Robustnessofinterdependentnetworksundertargetedattack.Phys.Rev.E83(2011):065101. [33] Hwang,W.,Cho,andRamanathan,M.BridgingCentrality:IdentifyingBridgingNodesinScale-freeNetworks.TechnicalReport,DepartmentofCSE,UniversityatBuffalo.2006. [34] Kempe,David,Kleinberg,Jon,andTardos,Eva.Maximizingthespreadofinuencethroughasocialnetwork.ProceedingsoftheninthACMSIGKDDinternationalconferenceonKnowledgediscoveryanddatamining.KDD'03.NewYork,NY,USA:ACM,2003,137. [35] .Inuentialnodesinadiffusionmodelforsocialnetworks.Proceedingsofthe32ndinternationalconferenceonAutomata,LanguagesandProgramming.ICALP'05.Berlin,Heidelberg:Springer-Verlag,2005,1127. [36] Kinney,Reka,Crucitti,Paolo,Albert,Reka,andLatora,Vito.ModelingcascadingfailuresintheNorthAmericanpowergrid.TheEuropeanPhysicalJournalB-CondensedMatterandComplexSystems46(2005).1:101. [37] Leskovec,Jure,Krause,Andreas,Guestrin,Carlos,Faloutsos,Christos,VanBriesen,Jeanne,andGlance,Natalie.Cost-effectiveoutbreakdetectioninnetworks.Proceedingsofthe13thACMSIGKDDinternationalconferenceonKnowledgediscoveryanddatamining.KDD'07.NewYork,NY,USA:ACM,2007,420. [38] Liu,Xingjie,He,Qi,Tian,Yuanyuan,Lee,Wang-Chien,McPherson,John,andHan,Jiawei.Event-basedsocialnetworks:linkingtheonlineandofinesocialworlds.Proceedingsofthe18thACMSIGKDDinternationalconferenceonKnowledgediscoveryanddatamining.KDD'12.NewYork,NY,USA:ACM,2012,1032. 100

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[53] Wang,Jian-WeiandRong,Li-Li.Cascade-basedattackvulnerabilityontheUSpowergrid.SafetyScience47(2009).10:13321336. [54] Wang,Wenkai,Cai,Qiao,Sun,Yan,andHe,Haibo.Risk-awareattacksandcatastrophiccascadingfailuresinuspowergrid.GlobalTelecommunicationsConference(GLOBECOM2011),2011IEEE.IEEE,2011,1. [55] Watts,DuncanJandStrogatz,StevenH.Collectivedynamicsofsmall-worldnetworks.nature393(1998).6684:440. [56] Wu,Zhi-Xi,Peng,Gang,Wang,Wen-Xu,Chan,Sammy,andWong,EricWing-Ming.Cascadingfailurespreadingonweightedheterogeneousnetworks.JournalofStatisticalMechanics:TheoryandExperiment2008(2008).05:P05013. [57] Xia,Yongxiang,Tse,ChiK.,Tam,WaiM.,Lau,FrancisC.M.,andSmall,Michael.Scale-freeuser-networkapproachtotelephonenetworktrafcanalysis.Phys.Rev.E72(2005):026116. [58] Yagan,Osman,Qian,Dajun,Zhang,Junshan,andCochran,Douglas.Informationdiffusioninoverlayingsocial-physicalnetworks.InformationSciencesandSystems(CISS),201246thAnnualConferenceon.2012,1. [59] Zhao,Liang,Park,Kwangho,andLai,Ying-Cheng.Attackvulnerabilityofscale-freenetworksduetocascadingbreakdown.PhysicalreviewE70(2004).3:035101. [60] Zhao,Liang,Park,Kwangho,Lai,Ying-Cheng,andYe,Nong.Toleranceofscale-freenetworksagainstattack-inducedcascades.PhysicalReviewE72(2005).2:025104. [61] Zhu,Xu,Yu,Jieun,Lee,Wonjun,Kim,Donghyun,Shan,Shan,andDu,Ding-Zhu.Newdominatingsetsinsocialnetworks.JournalofGlobalOptimization48(2010).4:633. [62] Zou,Feng,Zhang,Zhao,andWu,Weili.Latency-BoundedMinimumInuentialNodeSelectioninSocialNetworks.WASA.2009,519. 102

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BIOGRAPHICALSKETCH DungT.NguyenreceivedtheBSdegreeinInformationTechnologyfromHanoiUniversityofScienceandTechnology,Hanoi,Vietnamin2008.HeiscurrentlyaPhDstudentattheDepartmentofComputerandInformationScienceandEngineering,UniversityofFlorida,underthesupervisionofDr.MyT.Thai.Hisareasofinterestareviralmarketingononlinesocialnetworks,vulnerabilityandcascadingfailuresoncouplednetworks,andapproximationalgorithmsfornetworkoptimizationproblems. 103