The resilience of networks to various types of failures is an undercurrent in many parts of graph theory and network algorithms In this paper we study the resilience of networks in the presence of cascading failures failures that spread from one no ID: 27222
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1IntroductionTheresilienceofnetworkstovarioustypesoffailuresisanundercurrentinmanypartsofgraphtheoryandnetworkalgorithms.Forexample,thedenitionsofcutsandexpansioneachcapturetypesofrobustnessinthepresenceofworst-caseedgeornodedeletion,whilethestudyofnetworkreliabilityisbasedonthequestionofconnectivityinthepresenceofprobabilisticedgefailures,amongotherissues.Inthispaperweareinterestedintheresilienceofnetworksinthepresenceofcascadingfailures|failuresthatspreadfromonenodetoanotheracrossthenetworkstructure.Onendssuchcascadingprocessesatworkinthekindofcontagiousfailuresthatspreadamongnancialinstitu-tionsduringanancialcrisis[1],inthebreakdownsthatspreadthroughnodesofapowergridorcommunicationnetworkduringawidespreadoutage[3],orinthecourseofanepidemicdiseaseasitspreadsthroughahumanpopulation[2].Torepresentcascadingfailuresweusethefollowingbasicthresholdcascademodel,whichhasbeenstudiedextensivelybothinthecontextoffailuresandalsoinothersettingsinvolvingsocialorbiologicalcontagion[6,8,9,10,11,12,13,14].1WearegivenagraphG,andeachnodevchoosesathreshold`(v)independentlyfromadistributiononthenaturalnumbers,choosingthreshold`(v)=jwithprobability(j).Thequantity`(v)representsthenumberoffailedneighborsthatvcanwithstandbeforevfailsaswell|thuswecanthinkofasdeterminingthedistributionoflevelsof\health"ofthenodesinthepopulation,andhenceimplicitlycontrollingthewaythefailureprocessspreadsonG.Todeterminetheoutcomeofthefailureprocess,werstdeclareallnodeswiththreshold0tohavefailed.Wethenrepeatedlycheckwhetheranynodevthathasnotyetfailedhasatleast`(v)failedneighbors|ifso,wedeclarevtohavefailedaswell,andwecontinueiterating.Forexample,Figure1showstheoutcomeofthisprocessontwodierentgraphsGwithparticularchoicesofnodethresholds.ForagivennoderinG,wedeneitsfailureprobabilityf(G;r)tobetheprobabilityitfailswhennodethresholds`(v)aredrawnindependentlyfromandthenthethresholdcascademodelisrunwiththesethresholds.Nowweletf(G)=supr2V(G)f(G;r),namely,themaximumfailureprobabilityinG.Weviewf(G)asourmeasureoftheresilienceofGagainstcascadingfailuresthatoperateunderthethresholddistribution;accordingly,werefertof(G)asthe-riskofG,andweseekgraphsoflow-risk.AMotivatingContrast:CliquesandTrees.Howdodierentnetworkstructurescompareintheirresiliencetoacascadingfailure?Becausethefailureprobabilityclearlygoesupasweaddedgestoagivennodeset,wetakethetop-levelissueofedgedensityoutofconsiderationbyposingthisquestionoverthesetofall(niteorinnite)connectedd-regulargraphs,foraxedchoiceofd.WeuseGdtodenotethissetofgraphs,andforgraphsinGdweaskhowtheycompareaccordingtotheir-risk.2WhenweconsiderGd,wewillalsorestrictthethresholddistributionstothesetofalldistributionssupportedonf0;1;2;:::;dg,asetwhichwedenotebyd.Asarstconcreteexampleofthekindofresultstocome,weconsideracomparisonbetweentwobasicd-regulargraphs;theanalysisjustifyingthiscomparisonwillfollowfromtheframework 1Thethresholdcascademodelisalsorelatedtothenonlinearvotermodel[7],thoughsomewhatdierentinitsspecics.2Unlessexplicitlynotedotherwise,allquanticationovergraphsinthispapertakesplaceoverthesetofconnectedgraphsonly.Thisdoesnotcomeatanyreallossofgenerality,sincethe-riskofadisconnectedgraphissimplythesupremumofthe-riskineachconnectedcomponent.2 T2,theinnite2-arytree,isbetterknownastheinnitepath).Wendinfactthatateachwith0(0)1,atleastoneofK3orT2achievesstrictlylower-riskthaneveryothergraphinG2fK3;T2g.Whend]TJ/;ø 1;.90; T; 11;.515; 0 T; [0;2,thebehaviorof-riskonGdbecomesmuchmorecomplicated.Hereweestablishthatforeachd]TJ/;ø 1;.90; T; 11;.515; 0 T; [0;2,thetwographsfKd+1;Tdgdonotformasucientsetford.Wedothisbyconsideringagraphthatwecallthe(d-regular)treeoftrianglesd,consistingessentiallyofacollectionofdisjointtrianglesattachedaccordingtothestructureofaninniteregulartree.(disspeciedpreciselyinSection5.2,anddepictedschematicallyforthecased=3inFigure2).Weconstructadistribution2dforwhichdhasstrictlylower-riskthanbothKd+1andTd.Intuitively,thetreeoftriangles\interpolates"betweenthecompleteneighborhooddiversicationofTdandthecompleteneighborhoodclosureofKd+1,andhencepointstowardafurtherstructuraldimensiontotheproblemofminimizing-risk.Despitethecomplexstructureof-riskwhend]TJ/;ø 1;.90; T; 11;.515; 0 T; [0;2,wehaveasetofresultsmakingitpossibletocomparethe-riskofcertainspecicgraphstothe-riskofarbitrarygraphs.InadditiontothecomparisonsamongKd+1,Td,andddescribedabove,weestablishthefollowingfurtherresultsforKd+1andTd.First,asnotedabove,itisnothardtoshowthattherearedistributions2dforwhichKd+1hasstrictlylower-riskthananyotherG2Gd.AmuchmoreintricateargumentestablishesatypeofoptimalitypropertyforTdaswell:foreachgraphG2Gd,weconstructadistributionG2dforwhichTdhasstrictlylowerG-riskthanG.ThisisabroadgeneralizationoftheTd-vs.-Kd+1comparison,inthatitsaysthatsuchacomparisonispossibleforeveryG2Gd:inotherwords,Tdismoreresilientthaneveryotherconnectedd-regulargraphatsomepointind.OuranalysisinfactestablishesastrengtheningofthisresultforTd|foreverynitesetHofconnectedd-regulargraphs,thereisadistributionH2donwhichTdachievesstrictlylowerH-riskthaneachmemberofH.AndthisinturnyieldsanegativeanswertoamoregeneralversionofQuestion():Whend]TJ/;ø 1;.90; T; 11;.515; 0 T; [0;2,thereisnotwo-elementsucientsetofgraphsford.Ourresultsford]TJ/;ø 1;.90; T; 11;.515; 0 T; [0;2arebasedonaunifyingtechnique,motivatedbytheconstructionofthedistribution=(";x;1"x)usedtocompareKd+1andTdabove.Thetechniqueisbasedonusingpowerseriesapproximationstostudythe-riskforinthevicinityofparticularthresholddistributions;roughlyspeaking,itworksasfollows.Wefocusoncasesinwhichthedistributionconcentratesalmostallofitsprobabilityonasinglethreshold`maxandtheremainingprobabilityisdividedupovervaluesj`max.Therandomdrawofathresholdfrominthiscasecanbetreatedasasmallperturbationofthexedthresholddistributioninwhicheverynodegetsthreshold`maxandnonodesfail.Agivennode'sfailureprobabilitycanthenbeexpressedusingapowerseriesinthevariablesf(j)jj`maxgandthepowerseriescoecientsfordierentgraphsprovideenoughinformationtocomparethemaccordingto-riskwhentheprobabilitiesf(j)jj`maxgaresucientlyclosetozero.ThecomputationofthepowerseriescoecientsthenreducestoacountingprobleminvolvingcertainpartialassignmentsofthresholdstonodesofG.Inadditiontotheirroleinouranalyses,webelievethatsmallperturbationsofasinglexedthresholdareaverynaturalspecialcasetoconsiderforthethresholdcascademodel.Specically,lethd(x)dbethesetofdistributionsindsuchthat(0)-298;0,(j)xforjh,and(j)=0forj-278;h.(Inotherwords,mostoftheprobabilitymassisconcentratedonh,andtherestisonvaluesbelowh.)Thresholddistributionsinhd(x)forsmallx-278;0correspondtoscenariosinwhichallnodesbeginwithaxedlevelof\health"h,andthenashocktothesystemcausesasmallfractionofnodestofail,andasmallfractionofotherstobeweakened,withpositivethresholds5 `=0.If`(r)=2,output\fail"ifandonlyif`+=0and`=0.Denethelengthofanexecutionofthisalgorithmtobeequaltoi+j.(Notethatifi=1orj=1,thealgorithmRootFailwillnotactuallyhalt.Forthisreason,anactualimplementationofRootFailwouldhavetobemorecarefultoinspecttheverticesininterleavedorder|R(r);R1(r);R2(r);R2(r);:::|untilitcanprovethattherootmustfail.Suchanimplementationisnotguaranteedtohalt,butwhenprocessinganylabeling`suchthatr2(G;`)itisguaranteedtohaltafteranitenumberofstepsandoutput\fail".)Thekeytoanalyzingtherootfailureprobabilityin2-regulargraphsisthefollowingobservation:thereisaprobabilisticcouplingofthelabelings`PoftheinnitepathPandthelabelings`Cofthen-cycleC=Cn,suchthatforeverysamplepointatwhichRootFail(P;`P)hasexecutionlengthlessthann,RootFail(C;`C)alsohasexecutionlengthlessthannandthetwoexecutionsareidentical.Wenowdenesomeeventsonthesamplespaceofthiscoupling.Foranyk,letEkdenotetheeventthatRootFail(P;`P)hasexecutionlengthatleastk.LetFPdenotetheeventthatr2(P;`P)andletFCdenotetheeventthatr2(C;`C).SincetheexecutionsofRootFail(P;`P)andRootFail(C;`C)areidenticalonthecomplementofEn,wendthatPr(FP)Pr(FC)=Pr(En)[Pr(FPjEn)Pr(FCjEn)]:Wenowproceedtocomputeeachoftheconditionalprobabilitiesontheright-handside.Lets;t;udenotethelabelprobabilities(0);(1);(2);respectively.Letq=s 1t,whichistheconditionalprobabilitythatthelabelofanynodeis0,giventhatitslabelisnot1.ThenwehavePr(FPjEn)=t t+u1(1q)2+u t+uq2:Thersttermontherightaccountsforthecasethat`(r)=1andthesecondtermaccountsforthecasethat`(r)=2.Aftersomemanipulation|pullingoutt t+uqfromthersttermandu t+uqfromthesecondone|weobtaintheformulaPr(FPjEn)=q+tu t+uqq2:TocomputePr(FCjEn),notethatwhenEnoccurs,theroot'slabeliseither1or2,andatmostoneoftheremaininglabelsisnotequalto1.Furthermore,inanysuchlabelingofC,therootfailsifandonlyifoneoftheothern1nodeshaslabel0.Thus,Pr(En)=(t+u)[tn1+(n1)(1t)tn2]Pr(En\FC)=(t+u)(n1)stn2Pr(FCjEn)=(n1)s t+(n1)(1t)=q1t t+(n1)(1t)Pr(FPjEn)Pr(FCjEn)=tu t+uqq2+qt t+(n1)(1t)Pr(FP)Pr(FC)=Pr(En)tu t+uqq2+qt t+(n1)(1t):Onthelastline,bothfactorsaredecreasingfunctionsofn.Consequently,whentheyarebothpositive,theirproductisadecreasingfunctionofn.Inotherwords,ifann-cycleisbetterthananinnitepath,thenan(n1)-cycleisbetterstill.Wehavethusprovedthefollowing.7 3s2t+s2t=2s2ttotheprobability,withthersttermcomingfromtwo-wayoverlapsandthesecondtermcomingfromthethree-wayoverlap.Puttingallthistogether,wegettherootfailureprobabilityforthesmallexample:s+2st+s2s32s2t.MERFSgiverisetosuchoverlapswhentheyarecompatible.Wesatthattwopartiallabelings1;2arecompatibleif1(v)=2(v)foreveryv2Dom(1)\Dom(2).Theunionoftwocompatiblepartiallabelings1;2istheuniquepartialfunctionsuchthatf(v;(v))jv2Dom()g=f(v;1(v))jv2Dom(1)g[f(v;(v))jv2Dom(2)g:Fornotationalreasons,itwillbeconvenienttomaketheunionoperationintoabinaryoperationthatisdenedforanypairofpartiallabelings,notonlyforcompatiblepairs.Todoso,wedenethesettobeasetconsistingofallpartiallabelings,togetherwithonespecialelementdenoted?thatisinterpretedtobeincompatiblewitheveryelementof,includingitself.Weextendtheunionoperation[toabinaryoperationonbyspecifyingthat1[2=?when1and2areincompatible.Forapartiallabeling,wedeneE()tobethesetofallfulllabelingsthatextend;notethatE(?)=;,andthatforeverytwopartiallabelings1;2wehavetherelationE(1)\E(2)=E(1[2):Fortheinclusion-exclusionformula,we'llneedtothinkaboutniteunionsofMERFswhichwe'llcallUMERFs.ForgraphGwithrootvertexr,wewilldenotethesetofallMERFsbyM(G;r)andthesetofallUMERFsbyU(G;r).WewillsometimesabbreviatethesetoM;Uwhentheidentityofthegraphandrootvertexareobviousfromcontext.Wecannowdescribetheplanforarbitrarygraphs,includinginniteones,when(j)=sjaresmallnumbersforj`max,and(`max)=1P`max1j=0sj.Werstshowthatwhen`max-317;d=2,foranyvectorofnaturalnumbersi=(i0;i1;:::;i`max1),thereareonlynitelymanyMERFsthatassigniknodesalabelofk,fork=0;:::;`max1.Moreover,wecanwritetheroot'sfailureprobabilityasamultivariatepowerseriesoftheformPiaisi00si11si`max1`max1,andthispowerserieshasapositiveradiusofconvergence.Weusethistocomparefailureprobabilitiesindierentgraphsbyenumeratinganitesetoftermsinthepowerseriesuntilweidentifyadierencebetweenthem.4.2ApowerseriesforcomputingtherootfailureprobabilityWemakethesetofalllabelings`intoaprobabilityspacebydeclaringthelabelsf`(v)jv2V(G)gtobeindependentrandomvariableswithcommondistribution.Themeasurablesetsinthisprobabilityspacearethe-eldgeneratedbythesetsE(),whererangesoverallpartiallabelingsofG.ByLemma2.1,whenevertherootfailsthereisaMERFthatexplainsthefailure,i.e.theeventr2(G;`)istheunionoftheeventsE()for2M.SinceMisacountableset,wecanchooseanarbitraryone-to-onecorrespondencem:N!M.ThenPr(r2(G;`))=Pr 1[i=1E(m(i))!=limn!1Pr n[i=1E(m(i))!:(1)Eachoftheprobabilitiesontheright-handsidecanbeexpandedusingtheinclusion-exclusion9 1.IfGisd-regularandd2`maxthenjFj(d+1)jj.2.SupposethatforeverynodevofG,everyconnectedcomponentofGnfvgcontainsstrictlyfewerthan`maxneighborsofv.ThenjFj2jj.Proof.ArrangetheelementsofFintoasequencev1;v2;:::suchthateachofthesetsiG; isaninitialsegmentofthesequence.Thus,eachv2Fhasatleast (v)neighborsthatprecedeitinthesequence.Wecanthinkofthesequencev1;v2;:::asspecifyingapossibleorderinwhichthenodesofFfailedinanexecutionofthethresholdcascademodel.Toprovebothpartsofthelemmawewilldeneapotentialfunctionthatmapsvertexsetstonon-negativeintegers,thenevaluatethepotentialfunctiononeachinitialsegmentofthesequence,andconsiderhowthevalueofthepotentialfunctionchangeseverytimeanewnodefails(i.e.,isaddedtotheinitialsegment).Wewillusetwodierentpotentialfunctionscorrespondingtothetwoparts.ForPart1dene'(S),foranyvertexsetS,tobethenumberofedgesofGhavingoneendpointinSandtheotherinitscomplement.Eachtimeanewnodevkfails,itincreasesthevalueof'byatmostdsinceithasonlydneighbors.Furthermore,ifvk62Dom()thenvkhasatleast`maxneighborsthatprecedeitinthesequenceandatmostd`maxthatsucceedit.Thus,thenetchangein'isboundedaboveby(d`max)`max,whichisatmost1byourassumptionthatd2`max.Thepotentialfunction'thusstartsat0,increasesbyatmostdjjoverthewholesequenceoffailures,andisnevernegative;hencetherecanbeatmostdjjstepsofthesequencewhenitstrictlydecreases,andthereforeatmostdjjnodesinFnDom().ConsequentlyjFj(d+1)jj.ForPart2,weinsteadusethepotentialfunction (S)denedasthenumberofconnectedcomponentsintheinducedsubgraphG[S].Eachtimeanewnodevkfails,itincreases byatmost1.Nowconsiderhow changeswhenanodew62Dom()fails.Since (w)=`max,weknowthatwhasatleast`maxneighborsthatprecedeitinthesequence.ByourassumptiononthestructureofG,atleasttwooftheseneighborsbelongtodierentconnectedcomponentsofGnfwg.Thesecomponentsmergetogetherwhenwfails,causing todecreasebyatleast1.Sincetheinitialvalueof is0anditsnalvalueisstrictlypositive,anditincreasesbyatmost1ineachstep,weknowthatthenumberofstepsinwhich increasesmustbegreaterthanthenumberofstepsinwhichitdecreases.Hence,jFnDom()jjj,implyingjFj2jjasclaimed. ThenextlemmaprovidesasimplemethodforboundingthenumberofUMERFsofsizezbyanexponentialfunctionofz.Lemma4.3.Suppose,foragivengraphGanddefaultthreshold`max,thatthereexistsaconstantcsuchthateverypartiallabelingsatisesjG; jcjj.Thenforeveryz,thenumberofUMERFsofsizezisatmost(d+1)3cz.Inparticular,thisupperboundisatmost(d+1)3(d+1)zwheneveroneofthesucientconditionsinLemma4.2holds.Proof.LetbeapartiallabelingandletF=G; .IfisaMERF,thenFinducesaconnectedsubsetofG,sinceotherwisewecouldremovethelabelsprovidedbyinanycomponentofG[F]notcontainingtherootrandarriveatapropersublabelingofthatisalsoanERF.ThisimpliesthatifisaUMERF,thesetG[F]mustalsobeconnected,sinceitistheunionofanitesetofconnectedgraphsallcontainingacommonnoder.WecandescribeanysuchFuniquelybyspecifyingthesequenceofedgelabels(eachindexedfrom1tod)thataretakeninthe2jFjstepsofadepth-rstsearchtraversalofG[F]startingfromr.Hencethereareatmostd2jFjd2cjjsuchsets.AseachUMERFofsizejjisuniquelyassociatedwithsuchasetFtogetherwithalabeling11 Proof.Weprovethisbyinductiononn,withthecaseofn=1beingeasy.Forn1,chooseanyelementx2DandletD0=Dx.WedeneC0tobethecollectionofallJf1;:::;mgforwhichSj2JDjD0.andC1tobethecollectionofallJf1;:::;mgforwhichSj2JDj=D0.Now,bytheinductionhypothesisappliedtothesetsD0andfDjx:j=1;2;:::;mg,wehavePJ2C0(1)jJj2n1.BytheinductionhypothesisappliedtothesetsD0andfDj:x62Djg,wehavePJ2C1(1)jJj2n1.Finally,C1C0andJ2CifandonlyifJ2C0C1,sowehavePJ2C(1)jJj=PJ2C0(1)jJjPJ2C1(1)jJj,fromwhichitfollowsthatPJ2C(1)jJj2n. Puttingtheseboundstogether,weseethatjaijisboundedabovebyanexponentialfunctionofjij,andhence:Theorem4.5.Ifd2`max,thepowerseriesinEquation(5)hasapositiveradiusofconvergence.Thepowerseriesalsohasapositiveradiusofconvergenceifforeverynodev,everyconnectedcomponentofGnfvgcontainsstrictlyfewerthan`maxneighborsofv.5ComparingCliques,Trees,andTreesofTriangles5.1ComparingTdtoKd+1Intheintroduction,wenotedthatitiseasytoidentifytwodistinctsettingsoftheparametersforforwhichKd+1hasuniquelyoptimal-riskamongconnectedd-regulargraphs.First,when`max=1,theprobabilitytherootfailsismonotonicinthesizeoftheconnectedcomponentthatcontainsit,andKd+1uniquelyminimizesthisforconnectedd-regulargraphs.ButKd+1isalsouniquelyoptimalforlargervaluesof`maxd,whenassignseverylabeltobeeither0or`max.Indeed,inthiscase,theonlywaytherootcanfailinKd+1isifatleast`maxofitsneighborsfail.Thiseventalsocausestheroottofailinanyconnectedd-regulargraphG,butwhenG6=Kd+1thereareotherpositive-probabilityeventsthatalsocausetheroottofail,soagainKd+1isuniquelyoptimal.Asarstapplicationofourpower-seriestechnique,wenowshowthatthereareparametersettingsforwhichTdhaslowerrootfailureprobabilitythanKd+1.Forthiscomparison,weconsidersuchthat`max=2,andlabel0hasprobabilitys,whilelabel1hasprobabilityt,wheresandtaresmallquantitiesthatwillbedenedpreciselylater.Observethatwhen`max=2,TdsatisesthehypothesisofLemma4.2,Part2,andhenceitspowerserieshasapositiveradiusofconvergence.ThepowerseriesforKd+1isactuallyapolynomialinsandt,sinceKd+1isanitegraph,soitsradiusofconvergenceisinnite.Letusworkoutsomeofthelow-degreetermsforTdandforKd+1.ForTd,thecoecientonthetermsis1,correspondingtotheMERFinwhichtherootgetslabeled0.Thecoecientonthetermstisd,correspondingtoMERFsinwhichtherootgetslabeled1andanyoneoftheroot'sdneighborsgetslabeled0.Therearenoinclusion-exclusioncorrectionscontributingtoeitherofthesecoecients.ForKd+1,thecoecientonthetermsis1,asinTd,correspondingtotherootgettinglabeled0.However,thecoecientonthetermstisd2:therearedMERFsinwhichtherootgetslabeled1andanyoneoftheroot'sdneighborsgetslabeled0;therearealsod(d1)moreMERFsinwhichoneneighboroftherootgetslabeled0andanothergetslabeled1.Now,supposewesets=t3.ThenthepowerseriesfortherootfailureprobabilityinTdist3+dt4+O(t5),whereasthepowerseriesfortherootfailureprobabilityinK4ist3+d2t4+O(t5).13 withgirthgreaterthanj+2,impliesthateverynon-backtrackingwalkofjhopsterminatingatrisasimplepath,andthatthevertexsetsofallthesesimplepathsaredistinct.Consequently,inbothGandTthenumberofMERFssuchthati()=(1;j)isequaltothenumberofj-hopnon-backtrackingwalksinad-regulargraph,i.e.d(d1)j1.Whenj=L2,theset(G;)induceseitheratree(inwhichcase,bythesamereasoningasbefore,itmustbeaj-hoppathfromvtor)oranL-cyclecontainingasinglenodewsuchthat`(w)=2,asinglenodevwith`(v)=0,andallothernodeshavinglabel1.Inthelattercase,ourassumptionthateverynodeinDom()belongstoaMERFimplieseitherthatw=r,orthatv;w;roccurconsecutivelyonthecycleCandthatCnfwgisaj-hoppathfromvtor.Finally,itiseasytoseethatinallofthesecases,isaMERF.WehavethusshownthateveryUMERFinGwithi()=(1;L2)isaMERF,andthatthenumberoftheseMERFsisatleastd(d1)L3+L1.Here,d(d1)L3countsthenumberof(L2)-hoppathsterminatingattheroot|whichisalsothecoecientofstL2inthepowerseriesforTd|andL1countsthenumberofwaysoflabelingCnfrgwithasingle0andL21's. ProofofTheorem6.1,d=3case.Asd2`max,thepowerseriesforf(G;r)andf(T;r0)convergeforsucientlysmallsandt.Thusthedierencef(G;r)f(T;r0),maybeexpressedasPi=(i;j)(aGijaTij)sitjwhereaGijandaTijarethepowerseriescoecientsin(5)forGandT,respectively.GroupingthetermsintothosewithLi+j2L2andthosewithLi+j2L1,wendthattherstsetoftermsincludesonlypairs(i;j)suchthati=1;0jL2,andbyLemma6.2,XLi+j2L2(aGijaTij)sitj=(aG1;L2aT1;L2)stL2(L1)t2L2:(6)Recall,fromLemmas4.3and4.4,thatthenumberofUMERFssuchthati()=(i;j)isboundedaboveby(d+1)3(d+1)(i+j)andthatthecoecientPk(1)k+1akforeachofthemisboundedby2i+jinabsolutevalue.Thus,XLi+j2L1(aGijaTij)sitj1Xk=2L1XLi+j=k2i+j+1(d+1)3(d+1)(i+j)tLi+j1Xk=2L1k2k+1(d+1)3(d+1)ktk1Xk=2L1[4(d+1)3(d+1)t]k=4(d+1)3(d+1)2L1t2L1 14(d+1)3(d+1)t;wherethelastlineisjustiedaslongasthedenominatorisstrictlypositive.Bychoosingtsucientlysmall,wecanensurenotonlythatthedenominatorisstrictlypositivebutthatthequantityonthelastlineislessthant2L2.Then,thepositive(L1)t2L2contributionfromthelow-degreetermsinthepowerseriesmorethanosetsthepossiblynegativecontributionfromthehigh-degreeterms,andthisprovesthatf(G;r)f(T;r0),asclaimed.ProofofTheorem6.1,d3case.Whencomparingtheinnited-regulartreeTdagainstanotherconnectedd-regulargraphGwhend2`max,atrickyissuearisesbecausethepowerseriesforGneednotconverge.Recall,however,thatthepowerseriesforTdstillconverges.Thus,to16 Fromthisstrongerformoftheresult,weobtainthefollowingimmediateconsequence.Theorem6.4.Whend3,thereisnosucientsetofsize2ford.Proof.IfthereweresuchasetHGdofsize2,thenitwouldhavetocontainKd+1,sinceKd+1uniquelyminimizesthe-riskforsomedistributions2d.TheothergraphinHcan'tbeTd,sincebyTheorem5.2thereareforwhichdhasstrictlylower-riskthanbothKd+1andTd.ButiftheothergraphinHweresomeG6=Td,thenbyTheorem6.3wecouldndaforwhichTdhaslower-riskthanbothKd+1andG,andsothisisnotpossibleeither. References[1]FranklinAllenandDouglasM.Gale.Financialcontagion.JournalofPoliticalEconomy,108(1):1{33,February2000.[2]RoyM.AndersonandRobertM.May.InfectiousDiseasesofHumans.OxfordUniversityPress,1992.[3]ChaleeAsavathiratham,SandipRoy,BernardLesieutre,andGeorgeVerghese.Thein uencemodel.IEEEControlSystemsMagazine,21(6):52{64,December2001.[4]LarryBlume,DavidEasley,JonKleinberg,RobertKleinberg,andEvaTardos.Networkformationinthepresenceofcontagiousrisk.InProc.12thACMConferenceonElectronicCommerce,2011.[5]BelaBollobas.RandomGraphs.CambridgeUniversityPress,secondedition,2001.[6]DamonCentolaandMichaelMacy.Complexcontagionsandtheweaknessoflongties.Amer-icanJournalofSociology,113:702{734,2007.[7]J.T.CoxandRichardDurrett.Nonlinearvotermodels.InRichardDurrettandHarryKesten,editors,RandomWalks,BrownianMotion,andInteractingParticleSystems,pages189{202.Birkhauser,1991.[8]PeterDoddsandDuncanWatts.Universalbehaviorinageneralizedmodelofcontagion.PhysicalReviewLetters,92(218701),2004.[9]MarkGranovetter.Thresholdmodelsofcollectivebehavior.AmericanJournalofSociology,83:1420{1443,1978.[10]DavidKempe,JonKleinberg,andEvaTardos.Maximizingthespreadofin uenceinasocialnetwork.InProc.9thACMSIGKDDInternationalConferenceonKnowledgeDiscoveryandDataMining,pages137{146,2003.[11]StephenMorris.Contagion.ReviewofEconomicStudies,67:57{78,2000.[12]ElchananMosselandSebastienRoch.Onthesubmodularityofin uenceinsocialnetworks.InProc.39thACMSymposiumonTheoryofComputing,2007.[13]ThomasSchelling.MicromotivesandMacrobehavior.Norton,1978.[14]DuncanJ.Watts.Asimplemodelofglobalcascadesonrandomnetworks.Proc.Natl.Acad.Sci.USA,99(9):5766{5771,April2002.18 d2s2t:Thed2(d2)MERFsfromKd+1forthistermarenotpresentinTd,butthed2UMERFseachcontributing1are.d2s2u:Thereared2MERFswheretherootgetslabeled2andtwoneighborsarelabeled0.(Theotherd2(d2)MERFsfromKd+1,wheretwoneighborsoftherootarelabeled0andathirdislabeled2,arenotpresenthere.)d(d1)st2:Agrandchildoftherootgetslabel0,andtheparentofthischildtogetherwiththerootgetlabel1.0stu:ThereisnoMERForUMERFthatassignslabelsof0,1,and2tothreenodes.Ford,wethinkoftherootrashavingneighborsetZ=fv0;v1;v2;:::;vd1g,wheretheonlyedgeamongthenodesinZisbetweenv0andv1.WewillrefertoneighborsofZthatdonotbelongtothesetZ[frgasdepth-twoneighbors.Againthetermsoftotaldegreeupto2,aswellasthes3andsu2terms,arethesameasforKd+1,andbythesamearguments.Fortheotherdegree-3terms:(2(d2)d2)s2t:Thereared2UMERFseachcontributing1asinKd+1andTd,andtherearealso2(d2)MERFsinwhichoneofv0orv1islabeled0,theotherislabeled1,andaneighboroftherootinZfv0;v1gislabeled0.d2s2u:Wegetthed2MERFsthatwerepresentinTd,butnottheadditionald2(d2)thatwerepresentinKd+1.(d+1)(d2)st2:Thereare(d2)(d1)MERFswheretherootandanodew2Zfv0;v1garelabeled1,andoneofw'sdepth-twoneighborsislabeled0.Thereare2(d2)moreMERFswhere,fori2f0;1g,therootandviarelabeled1,andoneofvi'sdepth-twoneighborsislabeled0.2stu:Therootislabeled2,oneofv0orv1islabeled1,andtheotherislabeled0.ForoneofthesegraphsG,letfG(s;t;u)denotetherootfailureprobability,andletgG(s;t;u)denotethesumofalltermsinthepowerseriesoftotaldegree4andhigher.ThenwehavefKd+1(s;t;u)fd(s;t;u)=d2(d3)2(d2)d2s2t+d2(d1)d2s2u+d2(d2)(d+1)(d2)st2+d(d1)22stu+(gKd+1(s;t;u)gd(s;t;u))=d22(d2)s2t+d2(d2)s2u+d2(d+1)(d2)st2+d(d1)22stu+(gKd+1(s;t;u)gd(s;t;u))20