Model for cascading failures in complex networks Paolo - PDF document

Model for cascading failures in complex networks Paolo
Model for cascading failures in complex networks Paolo

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Paolo 73 95123 Catania Italy Dipartimento di Fisica e Astronomia Universita di Catania and INFN Sezione di Catania Via S So731a 64 95123 Catania Italy W3C and Laboratory for Computer Science Massachusetts Institute of Technology Cambridge Massachuse ID: 65647 Download Pdf

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ModelforcascadingfailuresincomplexnetworksPaoloCrucitti,VitoLatora,andMassimoMarchioriScuolaSuperiorediCatania,ViaS.Paolo73,95123Catania,ItalyDipartimentodiFisicaeAstronomia,UniversitadiCatania,andINFN,SezionediCatania,ViaS.So®a64,95123Catania,ItalyW3CandLaboratoryforComputerScience,MassachusettsInstituteofTechnology,Cambridge,Massachusetts02139,USAReceived16September2003;published29April2004Largebutrarecascadestriggeredbysmallinitialshocksarepresentinmostoftheinfrastructurenetworks.Herewepresentasimplemodelforcascadingfailuresbasedonthedynamicalredistributionofthe¯owonthenetwork.Weshowthatthebreakdownofasinglenodeissuf®cienttocollapsetheef®ciencyoftheentiresystemifthenodeisamongtheoneswithlargestload.Thisisparticularlyimportantforreal-worldnetworkswithahighlyhetereogeneousdistributionofloadsastheInternetandelectricalpowergrids.DOI:10.1103/PhysRevE.69.045104PACSnumber:89.75.Hc,89.20.Hh,89.30.g,89.75.FbCascadingfailuresarecommoninmostofthecomplexcommunicationand/ortransportationnetworksthatarethebasiccomponentsofourlivesandindustry.Infact,al-thoughmostfailuresemergeanddissolvelocally,largelyun-noticedbytherestoftheworld,afewtriggeravalanchemechanismsthatcanhavelargeeffectsovertheentirenet-CascadingfailurestakeplaceontheInternet,wheretraf®cisreroutedtobypassmalfunctioningrouters,eventuallylead-ingtoanavalancheofoverloadsonotherroutersthatarenotequippedtohandleextratraf®c.Theredistributionofthetraf®ccanresultinacongestionregimewithalargedropintheperformance.ForinstanceinOctober1986,duringthe®rstdocumentedInternetcongestioncollapse,thespeedoftheconnectionbetweentheLawrenceBerkeleyLaboratoryandtheUniversityofCaliforniaatBerkeley,twoplacesseparatedonlyby200m,droppedbyafactor100Cascadingfailuresalsotakeplaceinelectricalpowergrids.Infact,whenforanyreasonalinegoesdown,itspowerisautomaticallyshiftedtotheneighboringlines,whichinmostofthecasesareabletohandletheextraload.Sometimes,however,theselinesarealsooverloadedandmustredistributetheirincreasedloadtotheirneighbors.Thiseventuallyleadstoacascadeoffailures:alargenumberoftransmissionlinesareoverloadedandmalfunctionatthesametime.Thisisexactlywhathappenedon10August1996whena1300-mwelectricallineinsouthernOregonsaggedinthesummerheat,initiatingachainreactionthatcutpowertomorethan4millionpeoplein11WesternStates.Andprobablythisisalsowhathappenedon14August2003whenaninitialdisturbanceinOhiotriggeredthelargestblackoutintheU.S.'shistoryinwhichmillionsofpeopleremainedwithoutelectricityforaslongas15h.LargecascadingfailuresarealsopresentinsocialandeconomicsystemsHowisitpossiblethatasmallinitialshock,suchasthebreakdownofanInternetrouterorofanelectricalsubsta-tionorline,cantriggeravalanchesmechanismsaffectingaconsiderablefractionofthenetworkandcollapsingasystemthatinthepastwasproventobestablewithrespecttosimi-larshocks?Inthispaperweproposeasimplemodelforcascadingfailuresincomplexnetworks.Resistanceofnet-workstotheremovalofnodesorarcs,dueeithertorandombreakdownsortointentionalattacks,hasbeenstudiedin9±13.Suchstudieshavefocusedonlyonthepropertiesofthenetworkshowingthattheremovalofagroupofnodesaltogethercanhaveimportantconsequences.Hereweshowhowthebreakdownofasinglenodeissuf®-cienttocollapsetheentiresystemsimplybecauseofthedynamicsofredistributionof¯owsonthenetwork.Inourmodeleachnodeischaracterizedbyagivenhandlethetraf®c.Initiallythenetworkisinastationarystateinwhichtheateachnodeissmallerthanitscapacity.Thebreakdownofanodechangesthebalanceof¯owsandleadstoaredistributionofloadsoverothernodes.Ifthecapacityofthesenodescannothandletheextraloadthiswillberedistributedinturn,triggeringacascadeofover-loadfailuresandeventuallyalargedropinthenetworkper-formancesuchasthoseobservedinrealsystems,liketheInternetortheelectricalpowergrids.Themaindifferenceswithrespecttopreviousmodels14±16areasfollows.Overloadednodesarenotremovedfromthenetwork.Itisthecommunicationpassingthroughoverloadednodesthatwillgetworse,sothateventuallytheinformation/energywillavoidcongestednodes.Thedamagecausedbyacascadeisquanti®edintermsofthedecreaseinthenetwork,avariablede®nedinRef.Firstweintroducethemodelandthenweshowsomeapplicationstoarti®ciallycreatedtopologies,totheInternet,andtotheelectricalpowergridofthewesternUnitedStates.Werepresentagenericcommunicationand/ortransporta-tionnetworkasavalued,withtheInternetroutersorthesubstationsofanelectricalpowergridthetransmissionisdescribedbytheadjacencymatrix.Ifthereisanarcbetweennodeandnode,theentryisthevalue,anumberintherangeattachedtothearc;other-.Suchanumberisameasureoftheef®-ciencyinthecommunicationalongthearc.Forinstance,intheInternet,thesmalleris,thelongerittakestoexchangeaunitarypacketofinformationalongthearcbetweenInitially,attime0,weset1foralltheexistingarcs,meaningthatallthetransmissionlinesworkperfectlyandareequivalent.ThemodelwewillproposeconsistsofaruleforthetimeevolutionofthatmimicsthedynamicsofRAPIDCOMMUNICATIONSPHYSICALREVIEWE,045104/$22.502004TheAmericanPhysicalSociety ¯owredistributionfollowingthebreakdownofanode.Tode®nethenetworkef®ciencyweassumethatthecom-municationbetweenagenericcoupleofnodestakesthemostef®cientpathconnectingthem.Theef®ciencyofapathistheso-calledharmoniccomposition21±23oftheef®cienciesofthecomponentarcs.Byweindicatetheef®ciencyofthemostef®cientpathbetween.Matrixiscal-culatedbymeansofthealgorithmsusedinRef..Thentheaverageef®ciencyofthenetworkis andisusedasameasureoftheperformanceofatagivenloadL)onnodeattimeisthetotalnumberofmostef®cientpathspassingthroughattime.Eachnodeischaracterizedbyade®nedasthemaximumloadthatnodecanhandle.FollowingRef.weassumethecapacityofnodetobeproportionaltoitsinitialload1,2,...,1isthetoleranceparameterofthenetworkThisisarealisticassumptioninthedesignofaninfrastruc-turenetwork,sincethecapacitycannotbein®nitelylargebecauseitislimitedbythecost.Withsuchade®nitionofcapacity,thenetworkwehavecreatedisinastationarystateinwhichitoperateswithacertainef®ciency.Theinitialremovalofanode,simulatingthebreakdownofanIn-ternetrouterorofanelectricalsubstation,startsthedynam-icsofredistributionof¯owsonthenetwork.Infacttheremovalofanodechangesthemostef®cientpathsbetweennodesandconsequentlythedistributionoftheloads,creatingoverloadsonsomenodes.Ateachtimeweadoptthefol-lowingiterativerule: extendstoallthe®rstneighborsof.Inthiswayifatanodeiscongested,wereducetheef®ciencyofallthearcspassingthroughit,sothateventuallytheinformation/energywilltakealternativepathsthenewmostef®cientpaths.Thisisasofterand,forsomeapplications,amorerealisticsituationthantheoneconsideredinRef.inwhichtheoverloadednodesareremovedfromthenet-work.Ruleproducesadecreaseoftheef®ciencyoftheand,aswewillshowinthefollowing,insomecasesitcantriggeranavalanchemechanismcollapsingthewholesystem.Weillustratehowourmodelworksinpracticebyconsid-eringtwoarti®ciallycreatednetworktopologies:randomgraphsscale-freenetworks,i.e.,graphswithanalgebraicdistributionofdegree3generatedaccordingtotheBarabaInbothcaseswehaveconstructednetworkswith2000and10000.InFig.1wereportthetypicaltimeevolutionofthenetworkef®ciencyfortheBAscale-freenetwork.Thedynamicsofredistributionof¯owsistriggeredbytheremovalattime0ofanodechosenatrandom.Weshowtheresultsforthreevaluesofthetoleranceparameter,namely,1.3,1.05,1.01.Inthe®rstcasetheef®ciencyofthenetworkiscompletelyunaffectedbythefailureofthenode.Inthesecondcasethenetworkreachesastationarystatewithanef®ciencylowerthantheinitialone.Inthethirdcase,becauseofthelowertoleranceparameter,thecascadingfailurescollapsethesystem:thenetworkhaslost40%oftheinitialef®ciency.InFig.2wereportthe®nalvalueoftheef®ciency,i.e.,theef®ciencyafterthesystemhasrelaxedtoastationarystate,asafunctionofthetoleranceparameter.Wecon-siderboththeERrandomgraphandtheBAscale-freegraph.Moreover,weadopttwodifferenttriggeringstrategies:domremovalsload-basedremovals.Inthe®rstcasethenoderemovedinitiallyischosenatrandom:inthiswaywesimulatethebreakdownoftheaveragenodeofthenetwork.Inthesecondcasefullcirclestheremovednodeisaveryspecialonebecauseitistheonewiththelargestload.Bothfortherandomandforthescale-freenet-workweobserveadecreaseoftheef®ciencyforsmallvaluesofthetoleranceparameter,andthecollapseofthesystemforvaluessmallerthanacriticalvalue.ERrandomgraphsappeartobemoreresistanttocascadingfailuresthanBAscale-freegraphsasalsofoundinthemodelofRef..Inbothcasesthecollapsetransitionisalwayssharperforload-basedremovalsthanforrandomremovals,althoughthevaluesofcan¯uctuatefordifferentrealizations.FortheERrandomgraphsconsideredwehaveobtained0.002forrandomremovals,andforload-basedremovals.ForBAscale-freegraphs0.004forrandomremovals,and0.05forload-basedremovals.Theheterogeneityofthenetworkplaysanimportantroleinthenetworkstability.ERrandomgraphs FIG.1.CascadingfailureinaBAscale-freenetworkastrig-geredbytheinitialremovalofasinglenodechosenatrandom.Weplottheef®ciencyofthenetworkasafunctionofthetimeforthreevaluesofthetoleranceparameter.Thecurvescorrespondtoanaverageovertentriggers.RAPIDCOMMUNICATIONSCRUCITTI,LATORA,ANDMARCHIORIPHYSICALREVIEWE,045104 haveanexponentialloaddistributionwhileBAnetworksex-hibitapower-lawdistributioninthenodeload.Thismakesalargedifferencebetweenrandomremovalsandload-basedremovalsinBAscale-freenetworks.Infacttherearefewnodes,theoneswithextremelyhighinitialload,thatarefarmorelikelythantheothernodesthemostpartofthenodesofnetworktotriggercascades.Figure2showstheexistenceofalargeregioninthetoleranceparameter,1.3,wherescale-freenetworksarestablewithre-specttorandomremovalsandareunstablewithrespecttoload-basedremovals.If,forinstancethenodesworkwithatoleranceof30%abovethestandardload(1.3),thenet-workisingeneralverystabletoaninitialshockconsistinginthebreakdownofanode.Thismeansthatinmostofthecasesthefailureisperfectlytoleratedandreabsorbedbythesystem.However,thereisalwaysa®nite,althoughverysmall,probabilitythatthefailuretriggersanavalanchemechanism,collapsingthewholenetwork.AsexamplesfromtherealworldwestudyanetworkoftheInternetattheautonomoussystemlevel6474nodesand12567arcstakenfromRef.,andtheelectricalpowergridofthewesternUnitedStatesfrom4941and6592.AlthoughtheInter-netexhibitsapowerlawdegreedistributionasforBAscale-freenetworkswhiletheelectricalpowergridhasanexpo-nentialdegreedistributionasforERrandomgraphs,wehavecheckedthatboththenetworksconsideredareveryhetereogeneousfromthepointofviewoftheloadsonnodes.IntheinsetsofFig.3andFig.4wereport),thenumberofnodeswithaloadlargerthan,asafunctionof:thestraightlinesindicatethattheloaddistributionisconsistentwithapower-lawwithexponents,respectively,of1.80and1.75.Inthesame®gureswereportthevalueoftheef®ciencyafterthecascadetriggeredbyrandomfailuresandload-basedfailures.Duetothepresenceofafewnodeswithanex-tremelyhighinitialload,the®guresshowalargerangeofwherethenetworkisstableagainstrandomfailuresandisvulnerablewithrespecttothebreakdownofthemostloadednodes.Althoughthelattereventshaveaverylowprobability,theiroccurrencemaycollapsetheentiresystemswithalargeeffectonourlife.TheseresultsareapossibleexplanationofthemechanismproducingtheexperimentallyobservedInter-netcongestioncollapsesandthepowerblackouts.Asmallinitialshock,suchasthebreakdownofanInternetrouterorofanelectricalsubstationorline,maytriggeravalanchemechanismsaffectingaconsiderablefractionofanetworkthatforyearswasproventobestablewithrespecttosimilarshocks.Asanexample,iftheelectricpowergridofthewest-ernUnitedStatesofFig.4workswithatolerance,acaseinwhichthesystemisstablewithrespecttothefailureofmostofitsnodes,theremovalofaspecialnode,theonewithhighestinitialload,producesadropofofitsef®ciency. FIG.2.CascadingfailureinERrandomgraphsandscale-freenetworksastriggeredbytheremovalofanodechosenat,orbytheremovalofthenodewithlargestloadfullcircles.Wereportthe®nalafterthecascadethenetworkasafunctionofthetoleranceparameter.Boththenetworksconsideredhave2000and10000.Inthecasetriggeredbytheremovalofanodechosenatrandom,thecurvecorrespondstoanaverageovertentriggers. FIG.3.CascadingfailureintheInternet.Thenetworkconsid-eredistakenfromRef..Foreachvalueofwereporttheafterthecascadetriggeredbytheremovalofanodechosenatrandom,orbytheremovalofthenodewiththelargestloadfullcircles.Thecurvereportedforrandomremovalsisanaverageovertendifferentnodes.Intheinsetweplotthecumulativenodeloaddistribution. FIG.4.CascadingfailureintheelectricalpowergridofthewesternUnitedStatesfromRef..SameplotasinFig.3.RAPIDCOMMUNICATIONSMODELFORCASCADINGFAILURESINCOMPLEXNETWORKSPHYSICALREVIEWE,045104 Summingup,inthispaperwehaveintroducedasimplemodeltoexplainwhylargebutrarecascadetriggeredbysmallinitialshocksarepresentinmostofthecomplexcommunication/transportationnetworksthatarethebasiccomponentsofourlives.Themodelisbasedonadynamicalredistributionofthe¯owtriggeredbytheinitialbreakdownofacomponentofthesystem.Theresultsshowthatthebreakdownofasinglenodeissuf®cienttoaffecttheef®-ciencyofanetworkuptothecollapseoftheentiresystemifthenodeisamongtheoneswiththelargestload.Thisisparticularlyimportantfornetworkswithahighlyhetereo-geneousdistributionofnodeloadssuchasBAscale-freenetworks,butalsoreal-worldnetworkssuchastheInternetandelectricalpowergrids.Ourresultsshowthatitisonlythebreakdownofaselectedminorityofthenodesthatcantriggerthecollapseofthesystem.Itisalsotruethatforthemajorityofthenodesnothingharmfulhappens,whichleadsustotheerroneousbeliefthatourcommunication/transportationnetworksaresafe.Therefore,itshouldbead-visabletotakeintoproperaccount,inthedesignofanycomplexnetwork,thecascadingfailureseffectsanalyzedWethankD.J.WattsfortheU.S.power-griddatafromandA.Rapisardaforusefulcomments. S.N.DorogovtesevandJ.F.F.Mendes,EvolutionofNetworksOxfordUniversityPress,Oxford,2003S.H.Strogatz,Nature,268V.Jacobson,Comput.Commun.Rev.,314R.Guimera,A.Arenas,A.Dõaz-Guilera,andF.Giralt,Phys.Rev.E,026704B.A.Carreras,D.E.Newman,I.Dolrou,andA.B.Poole,inProceedingsofHawaiiInternationalConferenceonSystemSciences,Maui,Hawaii,2000M.L.Sachtjen,B.A.Carreras,andV.E.Lynch,Phys.Rev.E,4877J.GlanzandR.Perez-Pena,``90SecondsThatLeftTensofMillionsofPeopleintheDark,''NewYorkTimes,August26,D.J.Watts,Proc.Natl.Acad.Sci.U.S.A.,5766R.Albert,H.Jeong,andA.-L.Barabasi,Nature,542P.Holme,B.J.Kim,C.N.Yoon,andS.K.Han,Phys.Rev.E,056109P.Crucitti,V.Latora,M.Marchiori,andA.Rapisarda,Physica,622M.GirvanandM.E.J.Newman,Proc.Natl.Acad.Sci.U.S.A.,8271A.E.Motter,T.Nishikawa,andY.Lai,Phys.Rev.EA.E.MotterandY.Lai,Phys.Rev.E,065102Y.Moreno,R.Pastor-Satorras,A.Vazquez,andA.Vespignani,Europhys.Lett.,292Y.Moreno,J.B.Gomez,andA.F.Pacheco,Europhys.Lett.V.LatoraandM.Marchiori,Phys.Rev.Lett.,198701S.WassermanandK.Faust,SocialNetworksAnalysisbridgeUniversityPress,Cambridge,1994Themodelcanbeeasilygeneralizedtodirectedgraphs.Inthecaseofanunweightedgraphis1ifthereisanarcjoiningnodetonode,and0otherwise.J.Smith,Commun.ACM,1202R.Jain,TheArtofComputerSystemsPerformanceAnalysisWiley,NewYork,1991Theharmoniccompositionof,...,isde-®nedas(1/K.-I.Goh,B.Kahng,andD.Kim,Phys.Rev.Lett.,278701Anotherpossibilitywhichgivesthesameresultsistoset1,2,...,Asaninitialdamagetothenetworkhereweconsiderthere-movalofanode,althoughthedynamicsofthemodelcanbestartedbytheremovalofonearcifonewantstosimulate,forinstance,thebreakdownofalineinanelectricalpowergrid.P.ErdosandA.Renyi,Publ.Math.,290A.-L.BarabasiandR.Albert,Science,509Forload-basedremovalsinBAscale-freegraphswehavefoundthatisindependentofthesizeofthesystem.Infact,wehaveobtained0.05,1.30.05,1.280.05,re-spectively,for1500,2000,2500.Sucharesultissimilartothatwhichisobtainedinasimplermodel,the®ber-bundlemodel,studiedinRef.R.Pastor-Satorras,A.Vazquez,andA.Vespignani,Phys.Rev.,258701http://moat.nlanr.net/AS/Data/ASconnlist.20000102.946809601.D.J.WattsandS.H.Strogatz,Nature,440RAPIDCOMMUNICATIONSCRUCITTI,LATORA,ANDMARCHIORIPHYSICALREVIEWE,045104

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