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Which Networks Are Least Susceptible to Cascading Failures Larry Blume David Easley Jon Which Networks Are Least Susceptible to Cascading Failures Larry Blume David Easley Jon

Which Networks Are Least Susceptible to Cascading Failures Larry Blume David Easley Jon - PDF document

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Which Networks Are Least Susceptible to Cascading Failures Larry Blume David Easley Jon - PPT Presentation

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,thede nitionsofcutsandexpansioneachcapturetypesofrobustnessinthepresenceofworst-caseedgeornodedeletion,whilethestudyofnetworkreliabilityisbasedonthequestionofconnectivityinthepresenceofprobabilisticedgefailures,amongotherissues.Inthispaperweareinterestedintheresilienceofnetworksinthepresenceofcascadingfailures|failuresthatspreadfromonenodetoanotheracrossthenetworkstructure.One ndssuchcascadingprocessesatworkinthekindofcontagiousfailuresthatspreadamong nancialinstitu-tionsduringa nancialcrisis[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,we rstdeclareallnodeswiththreshold0tohavefailed.Wethenrepeatedlycheckwhetheranynodevthathasnotyetfailedhasatleast`(v)failedneighbors|ifso,wedeclarevtohavefailedaswell,andwecontinueiterating.Forexample,Figure1showstheoutcomeofthisprocessontwodi erentgraphsGwithparticularchoicesofnodethresholds.ForagivennoderinG,wede neitsfailureprobabilityf(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.Howdodi erentnetworkstructurescompareintheirresiliencetoacascadingfailure?Becausethefailureprobabilityclearlygoesupasweaddedgestoagivennodeset,wetakethetop-levelissueofedgedensityoutofconsiderationbyposingthisquestionoverthesetofall( niteorin nite)connectedd-regulargraphs,fora xedchoiceofd.WeuseGdtodenotethissetofgraphs,andforgraphsinGdweaskhowtheycompareaccordingtotheir-risk.2WhenweconsiderGd,wewillalsorestrictthethresholddistributionstothesetofalldistributionssupportedonf0;1;2;:::;dg,asetwhichwedenoteby�d.Asa rstconcreteexampleofthekindofresultstocome,weconsideracomparisonbetweentwobasicd-regulargraphs;theanalysisjustifyingthiscomparisonwillfollowfromtheframework 1Thethresholdcascademodelisalsorelatedtothenonlinearvotermodel[7],thoughsomewhatdi erentinitsspeci cs.2Unlessexplicitlynotedotherwise,allquanti cationovergraphsinthispapertakesplaceoverthesetofconnectedgraphsonly.Thisdoesnotcomeatanyreallossofgenerality,sincethe-riskofadisconnectedgraphissimplythesupremumofthe-riskineachconnectedcomponent.2 T2,thein nite2-arytree,isbetterknownasthein nitepath).We ndinfactthatateachwith0(0)1,atleastoneofK3orT2achievesstrictlylower-riskthaneveryothergraphinG2�fK3;T2g.Whend&#x]TJ/;ø 1;�.90;‘ T; 11;&#x.515;&#x 0 T; [0;2,thebehaviorof-riskonGdbecomesmuchmorecomplicated.Hereweestablishthatforeachd&#x]TJ/;ø 1;�.90;‘ T; 11;&#x.515;&#x 0 T; [0;2,thetwographsfKd+1;Tdgdonotformasucientsetfor�d.Wedothisbyconsideringagraphthatwecallthe(d-regular)treeoftrianglesd,consistingessentiallyofacollectionofdisjointtrianglesattachedaccordingtothestructureofanin niteregulartree.(disspeci edpreciselyinSection5.2,anddepictedschematicallyforthecased=3inFigure2).Weconstructadistribution2�dforwhichdhasstrictlylower-riskthanbothKd+1andTd.Intuitively,thetreeoftriangles\interpolates"betweenthecompleteneighborhooddiversi cationofTdandthecompleteneighborhoodclosureofKd+1,andhencepointstowardafurtherstructuraldimensiontotheproblemofminimizing-risk.Despitethecomplexstructureof-riskwhend&#x]TJ/;ø 1;�.90;‘ T; 11;&#x.515;&#x 0 T; [0;2,wehaveasetofresultsmakingitpossibletocomparethe-riskofcertainspeci cgraphstothe-riskofarbitrarygraphs.InadditiontothecomparisonsamongKd+1,Td,andddescribedabove,weestablishthefollowingfurtherresultsforKd+1andTd.First,asnotedabove,itisnothardtoshowthattherearedistributions2�dforwhichKd+1hasstrictlylower-riskthananyotherG2Gd.AmuchmoreintricateargumentestablishesatypeofoptimalitypropertyforTdaswell:foreachgraphG2Gd,weconstructadistributionG2�dforwhichTdhasstrictlylowerG-riskthanG.ThisisabroadgeneralizationoftheTd-vs.-Kd+1comparison,inthatitsaysthatsuchacomparisonispossibleforeveryG2Gd:inotherwords,Tdismoreresilientthaneveryotherconnectedd-regulargraphatsomepointin�d.OuranalysisinfactestablishesastrengtheningofthisresultforTd|forevery nitesetHofconnectedd-regulargraphs,thereisadistributionH2�donwhichTdachievesstrictlylowerH-riskthaneachmemberofH.AndthisinturnyieldsanegativeanswertoamoregeneralversionofQuestion():Whend&#x]TJ/;ø 1;�.90;‘ T; 11;&#x.515;&#x 0 T; [0;2,thereisnotwo-elementsucientsetofgraphsfor�d.Ourresultsford&#x]TJ/;ø 1;�.90;‘ T; 11;&#x.515;&#x 0 T; [0;2arebasedonaunifyingtechnique,motivatedbytheconstructionofthedistribution=(";x;1�"�x)usedtocompareKd+1andTdabove.Thetechniqueisbasedonusingpowerseriesapproximationstostudythe-riskforinthevicinityofparticularthresholddistributions;roughlyspeaking,itworksasfollows.Wefocusoncasesinwhichthedistributionconcentratesalmostallofitsprobabilityonasinglethreshold`maxandtheremainingprobabilityisdividedupovervaluesj`max.Therandomdrawofathresholdfrominthiscasecanbetreatedasasmallperturbationofthe xedthresholddistributioninwhicheverynodegetsthreshold`maxandnonodesfail.Agivennode'sfailureprobabilitycanthenbeexpressedusingapowerseriesinthevariablesf(j)jj`maxgandthepowerseriescoecientsfordi erentgraphsprovideenoughinformationtocomparethemaccordingto-riskwhentheprobabilitiesf(j)jj`maxgaresucientlyclosetozero.ThecomputationofthepowerseriescoecientsthenreducestoacountingprobleminvolvingcertainpartialassignmentsofthresholdstonodesofG.Inadditiontotheirroleinouranalyses,webelievethatsmallperturbationsofasingle xedthresholdareaverynaturalspecialcasetoconsiderforthethresholdcascademodel.Speci cally,let�hd(x)�dbethesetofdistributionsin�dsuchthat(0)&#x-298;0,(j)xforjh,and(j)=0forj&#x-278;h.(Inotherwords,mostoftheprobabilitymassisconcentratedonh,andtherestisonvaluesbelowh.)Thresholddistributionsin�hd(x)forsmallx&#x-278;0correspondtoscenariosinwhichallnodesbeginwitha xedlevelof\health"h,andthenashocktothesystemcausesasmallfractionofnodestofail,andasmallfractionofotherstobeweakened,withpositivethresholds5 `�=0.If`(r)=2,output\fail"ifandonlyif`+=0and`�=0.De nethelengthofanexecutionofthisalgorithmtobeequaltoi+j.(Notethatifi=1orj=1,thealgorithmRootFailwillnotactuallyhalt.Forthisreason,anactualimplementationofRootFailwouldhavetobemorecarefultoinspecttheverticesininterleavedorder|R(r);R�1(r);R2(r);R�2(r);:::|untilitcanprovethattherootmustfail.Suchanimplementationisnotguaranteedtohalt,butwhenprocessinganylabeling`suchthatr2(G;`)itisguaranteedtohaltaftera nitenumberofstepsandoutput\fail".)Thekeytoanalyzingtherootfailureprobabilityin2-regulargraphsisthefollowingobservation:thereisaprobabilisticcouplingofthelabelings`Pofthein nitepathPandthelabelings`Cofthen-cycleC=Cn,suchthatforeverysamplepointatwhichRootFail(P;`P)hasexecutionlengthlessthann,RootFail(C;`C)alsohasexecutionlengthlessthannandthetwoexecutionsareidentical.Wenowde nesomeeventsonthesamplespaceofthiscoupling.Foranyk,letEkdenotetheeventthatRootFail(P;`P)hasexecutionlengthatleastk.LetFPdenotetheeventthatr2(P;`P)andletFCdenotetheeventthatr2(C;`C).SincetheexecutionsofRootFail(P;`P)andRootFail(C;`C)areidenticalonthecomplementofEn,we ndthatPr(FP)�Pr(FC)=Pr(En)[Pr(FPjEn)�Pr(FCjEn)]:Wenowproceedtocomputeeachoftheconditionalprobabilitiesontheright-handside.Lets;t;udenotethelabelprobabilities(0);(1);(2);respectively.Letq=s 1�t,whichistheconditionalprobabilitythatthelabelofanynodeis0,giventhatitslabelisnot1.ThenwehavePr(FPjEn)=t t+u1�(1�q)2+u t+uq2:The rsttermontherightaccountsforthecasethat`(r)=1andthesecondtermaccountsforthecasethat`(r)=2.Aftersomemanipulation|pullingoutt t+uqfromthe rsttermandu t+uqfromthesecondone|weobtaintheformulaPr(FPjEn)=q+t�u t+u�q�q2:TocomputePr(FCjEn),notethatwhenEnoccurs,theroot'slabeliseither1or2,andatmostoneoftheremaininglabelsisnotequalto1.Furthermore,inanysuchlabelingofC,therootfailsifandonlyifoneoftheothern�1nodeshaslabel0.Thus,Pr(En)=(t+u)[tn�1+(n�1)(1�t)tn�2]Pr(En\FC)=(t+u)(n�1)stn�2Pr(FCjEn)=(n�1)s t+(n�1)(1�t)=q1�t t+(n�1)(1�t)Pr(FPjEn)�Pr(FCjEn)=t�u t+u�q�q2+qt t+(n�1)(1�t)Pr(FP)�Pr(FC)=Pr(En)t�u t+u�q�q2+qt t+(n�1)(1�t):Onthelastline,bothfactorsaredecreasingfunctionsofn.Consequently,whentheyarebothpositive,theirproductisadecreasingfunctionofn.Inotherwords,ifann-cycleisbetterthananin nitepath,thenan(n�1)-cycleisbetterstill.Wehavethusprovedthefollowing.7 �3s2t+s2t=�2s2ttotheprobability,withthe rsttermcomingfromtwo-wayoverlapsandthesecondtermcomingfromthethree-wayoverlap.Puttingallthistogether,wegettherootfailureprobabilityforthesmallexample:s+2st+s2�s3�2s2t.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,itwillbeconvenienttomaketheunionoperationintoabinaryoperationthatisde nedforanypairofpartiallabelings,notonlyforcompatiblepairs.Todoso,wede nethesettobeasetconsistingofallpartiallabelings,togetherwithonespecialelementdenoted?thatisinterpretedtobeincompatiblewitheveryelementof,includingitself.Weextendtheunionoperation[toabinaryoperationonbyspecifyingthat1[2=?when1and2areincompatible.Forapartiallabeling,wede neE()tobethesetofallfulllabelingsthatextend;notethatE(?)=;,andthatforeverytwopartiallabelings1;2wehavetherelationE(1)\E(2)=E(1[2):Fortheinclusion-exclusionformula,we'llneedtothinkabout niteunionsofMERFswhichwe'llcallUMERFs.ForgraphGwithrootvertexr,wewilldenotethesetofallMERFsbyM(G;r)andthesetofallUMERFsbyU(G;r).WewillsometimesabbreviatethesetoM;Uwhentheidentityofthegraphandrootvertexareobviousfromcontext.Wecannowdescribetheplanforarbitrarygraphs,includingin niteones,when(j)=sjaresmallnumbersforj`max,and(`max)=1�P`max�1j=0sj.We rstshowthatwhen`max&#x-317;d=2,foranyvectorofnaturalnumbersi=(i0;i1;:::;i`max�1),thereareonly nitelymanyMERFsthatassigniknodesalabelofk,fork=0;:::;`max�1.Moreover,wecanwritetheroot'sfailureprobabilityasamultivariatepowerseriesoftheformPiaisi00si11si`max�1`max�1,andthispowerserieshasapositiveradiusofconvergence.Weusethistocomparefailureprobabilitiesindi erentgraphsbyenumeratinga nitesetoftermsinthepowerseriesuntilweidentifyadi erencebetweenthem.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;:::suchthateachofthesetsi�G; isaninitialsegmentofthesequence.Thus,eachv2Fhasatleast (v)neighborsthatprecedeitinthesequence.Wecanthinkofthesequencev1;v2;:::asspecifyingapossibleorderinwhichthenodesofFfailedinanexecutionofthethresholdcascademodel.Toprovebothpartsofthelemmawewillde neapotentialfunctionthatmapsvertexsetstonon-negativeintegers,thenevaluatethepotentialfunctiononeachinitialsegmentofthesequence,andconsiderhowthevalueofthepotentialfunctionchangeseverytimeanewnodefails(i.e.,isaddedtotheinitialsegment).Wewillusetwodi erentpotentialfunctionscorrespondingtothetwoparts.ForPart1de ne'(S),foranyvertexsetS,tobethenumberofedgesofGhavingoneendpointinSandtheotherinitscomplement.Eachtimeanewnodevkfails,itincreasesthevalueof'byatmostdsinceithasonlydneighbors.Furthermore,ifvk62Dom()thenvkhasatleast`maxneighborsthatprecedeitinthesequenceandatmostd�`maxthatsucceedit.Thus,thenetchangein'isboundedaboveby(d�`max)�`max,whichisatmost�1byourassumptionthatd2`max.Thepotentialfunction'thusstartsat0,increasesbyatmostdjjoverthewholesequenceoffailures,andisnevernegative;hencetherecanbeatmostdjjstepsofthesequencewhenitstrictlydecreases,andthereforeatmostdjjnodesinFnDom().ConsequentlyjFj(d+1)jj.ForPart2,weinsteadusethepotentialfunction (S)de nedasthenumberofconnectedcomponentsintheinducedsubgraphG[S].Eachtimeanewnodevkfails,itincreases byatmost1.Nowconsiderhow changeswhenanodew62Dom()fails.Since (w)=`max,weknowthatwhasatleast`maxneighborsthatprecedeitinthesequence.ByourassumptiononthestructureofG,atleasttwooftheseneighborsbelongtodi erentconnectedcomponentsofGnfwg.Thesecomponentsmergetogetherwhenwfails,causing todecreasebyatleast1.Sincetheinitialvalueof is0andits nalvalueisstrictlypositive,anditincreasesbyatmost1ineachstep,weknowthatthenumberofstepsinwhich increasesmustbegreaterthanthenumberofstepsinwhichitdecreases.Hence,jFnDom()jjj,implyingjFj2jjasclaimed. ThenextlemmaprovidesasimplemethodforboundingthenumberofUMERFsofsizezbyanexponentialfunctionofz.Lemma4.3.Suppose,foragivengraphGanddefaultthreshold`max,thatthereexistsaconstantcsuchthateverypartiallabelingsatis esj�G; 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,sinceitistheunionofa nitesetofconnectedgraphsallcontainingacommonnoder.WecandescribeanysuchFuniquelybyspecifyingthesequenceofedgelabels(eachindexedfrom1tod)thataretakeninthe2jFjstepsofadepth- rstsearchtraversalofG[F]startingfromr.Hencethereareatmostd2jFjd2cjjsuchsets.AseachUMERFofsizejjisuniquelyassociatedwithsuchasetFtogetherwithalabeling11 Proof.Weprovethisbyinductiononn,withthecaseofn=1beingeasy.Forn�1,chooseanyelementx2DandletD0=D�x.Wede neC0tobethecollectionofallJf1;:::;mgforwhichSj2JDjD0.andC1tobethecollectionofallJf1;:::;mgforwhichSj2JDj=D0.Now,bytheinductionhypothesisappliedtothesetsD0andfDj�x:j=1;2;:::;mg,wehave PJ2C0(�1)jJj 2n�1.BytheinductionhypothesisappliedtothesetsD0andfDj:x62Djg,wehave PJ2C1(�1)jJj 2n�1.Finally,C1C0andJ2CifandonlyifJ2C0�C1,sowehavePJ2C(�1)jJj=PJ2C0(�1)jJj�PJ2C1(�1)jJj,fromwhichitfollowsthat PJ2C(�1)jJj 2n. 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.Asa rstapplicationofourpower-seriestechnique,wenowshowthatthereareparametersettingsforwhichTdhaslowerrootfailureprobabilitythanKd+1.Forthiscomparison,weconsidersuchthat`max=2,andlabel0hasprobabilitys,whilelabel1hasprobabilityt,wheresandtaresmallquantitiesthatwillbede nedpreciselylater.Observethatwhen`max=2,Tdsatis esthehypothesisofLemma4.2,Part2,andhenceitspowerserieshasapositiveradiusofconvergence.ThepowerseriesforKd+1isactuallyapolynomialinsandt,sinceKd+1isa nitegraph,soitsradiusofconvergenceisin nite.Letusworkoutsomeofthelow-degreetermsforTdandforKd+1.ForTd,thecoecientonthetermsis1,correspondingtotheMERFinwhichtherootgetslabeled0.Thecoecientonthetermstisd,correspondingtoMERFsinwhichtherootgetslabeled1andanyoneoftheroot'sdneighborsgetslabeled0.Therearenoinclusion-exclusioncorrectionscontributingtoeitherofthesecoecients.ForKd+1,thecoecientonthetermsis1,asinTd,correspondingtotherootgettinglabeled0.However,thecoecientonthetermstisd2:therearedMERFsinwhichtherootgetslabeled1andanyoneoftheroot'sdneighborsgetslabeled0;therearealsod(d�1)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(d�1)j�1.Whenj=L�2,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;L�2)isaMERF,andthatthenumberoftheseMERFsisatleastd(d�1)L�3+L�1.Here,d(d�1)L�3countsthenumberof(L�2)-hoppathsterminatingattheroot|whichisalsothecoecientofstL�2inthepowerseriesforTd|andL�1countsthenumberofwaysoflabelingCnfrgwithasingle0andL�21's. ProofofTheorem6.1,d=3case.Asd2`max,thepowerseriesforf(G;r)andf(T;r0)convergeforsucientlysmallsandt.Thusthedi erencef(G;r)�f(T;r0),maybeexpressedasPi=(i;j)(aGij�aTij)sitjwhereaGijandaTijarethepowerseriescoecientsin(5)forGandT,respectively.GroupingthetermsintothosewithLi+j2L�2andthosewithLi+j2L�1,we ndthatthe rstsetoftermsincludesonlypairs(i;j)suchthati=1;0jL�2,andbyLemma6.2,XLi+j2L�2(aGij�aTij)sitj=(aG1;L�2�aT1;L�2)stL�2(L�1)t2L�2:(6)Recall,fromLemmas4.3and4.4,thatthenumberofUMERFssuchthati()=(i;j)isboundedaboveby(d+1)3(d+1)(i+j)andthatthecoecientPk(�1)k+1akforeachofthemisboundedby2i+jinabsolutevalue.Thus, XLi+j2L�1(aGij�aTij)sitj 1Xk=2L�1XLi+j=k2i+j+1(d+1)3(d+1)(i+j)tLi+j1Xk=2L�1k2k+1(d+1)3(d+1)ktk1Xk=2L�1[4(d+1)3(d+1)t]k=�4(d+1)3(d+1)2L�1t2L�1 1�4(d+1)3(d+1)t;wherethelastlineisjusti edaslongasthedenominatorisstrictlypositive.Bychoosingtsucientlysmall,wecanensurenotonlythatthedenominatorisstrictlypositivebutthatthequantityonthelastlineislessthant2L�2.Then,thepositive(L�1)t2L�2contributionfromthelow-degreetermsinthepowerseriesmorethano setsthepossiblynegativecontributionfromthehigh-degreeterms,andthisprovesthatf(G;r)�f(T;r0),asclaimed.ProofofTheorem6.1,d�3case.Whencomparingthein nited-regulartreeTdagainstanotherconnectedd-regulargraphGwhend2`max,atrickyissuearisesbecausethepowerseriesforGneednotconverge.Recall,however,thatthepowerseriesforTdstillconverges.Thus,to16 Fromthisstrongerformoftheresult,weobtainthefollowingimmediateconsequence.Theorem6.4.Whend3,thereisnosucientsetofsize2for�d.Proof.IfthereweresuchasetHGdofsize2,thenitwouldhavetocontainKd+1,sinceKd+1uniquelyminimizesthe-riskforsomedistributions2�d.TheothergraphinHcan'tbeTd,sincebyTheorem5.2thereareforwhichdhasstrictlylower-riskthanbothKd+1andTd.ButiftheothergraphinHweresomeG6=Td,thenbyTheorem6.3wecould ndaforwhichTdhaslower-riskthanbothKd+1andG,andsothisisnotpossibleeither. 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