# Connected components in random graphs with given expected degree sequences Fan Chung y Linyuan Lu Abstract We consider a family of random graphs with a given expected degree sequence - PDF document

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#### Connected components in random graphs with given expected degree sequences Fan Chung y Linyuan Lu Abstract We consider a family of random graphs with a given expected degree sequence - Description

Each edge is chosen independently with probability propor tional to the product of the expected degrees of its endpoints We examine the distribution of the sizesvolumes of the connected components which turns out depending primarily on the average d ID: 7250 Download Pdf

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ConnectedcomponentsinrandomgraphswithgivenexpecteddegreesequencesFanChungLinyuanLuAbstractWeconsiderafamilyofrandomgraphswithagivenexpecteddegreesequence.Eachedgeischosenindependentlywithprobabilitypropor-tionaltotheproductoftheexpecteddegreesofitsendpoints.Weexaminethedistributionofthesizes/volumesoftheconnectedcomponentswhichturnsoutdependingprimarilyontheaveragedegreeandthesecond-orderaveragedegree.Heredenotestheweightedaverageofsquaresoftheexpecteddegrees.Forexample,weprovethatthegiantcomponentexistsiftheexpectedaveragedegreeisatleast1,andthereisnogiantcomponentiftheexpectedsecond-orderaveragedegreeisatmost1.Examplesaregiventoillustratethatbothboundsarebestpossible.1IntroductionTheprimarysubjectinthestudyofrandomgraphtheoryistheclassicalrandomgraphn;p),asintroducedbyErd}osandRenyiin1959[19].Inn;p),everypairofasetofverticesischosentobeanedgewithprobability.Suchrandomgraphsarefundamentalandusefulformodelingproblemsinmanyapplications.However,arandomgraphinn;p)hasthesameexpecteddegreeateveryvertexandthereforedoesnotcapturesomeofthemainbehaviorsofnumerousgraphsarisingfromtherealworld.Itisimperativetoconsideraversatileandgeneralizedversionofrandomgraphs.Inthispaper,weconsiderrandomgraphswithgivenexpecteddegreesequenceswhichincludeasspecialcasesboththeclassicalrandomgraphsandtherandomgraphswith\power-law"degreedistributions.Manyrealisticgraphssatisfythepower-law[1,2,3,7,8,12,13,20,21,25,26,36].Namely,thefractionofverticeswithdegree UniversityofCalifornia,SanDiegoResearchsupportedinpartbyNSFGrantDMS0100472 isproportionalto1forsomeconstant1.Althoughhereweconsiderrandomgraphswithgeneralexpecteddegreedistributions,specialemphasiswillbegiventosparsegraphs(withaveragedegreeasmallconstant)andtopowerlawgraphs(seeSection9).Themethodsandresultsthatwederiveindealingwithrandomgraphswithgivenexpecteddegreedistributionareusefulnotonlyformodelingandanalyzingrealisticgraphsbutalsoleadingtoimprovementsforsomeproblemsonclassicalrandomgraphsaswell[14,29].Weconsiderthefollowingclassofrandomgraphswithagivenexpecteddegreesequence).Thevertexisassignedvertexweight.Theedgesarechosenindependentlyandrandomlyaccordingtothevertexweightsasfollows.Theprobabilitythatthereisanedgebetweenandproportionaltotheproductwhereandarenotrequiredtobedistinct.Therearepossibleloopsatwithprobabilityproportionalto,i.e., andweassumemax(1)Thisassumptionensuresthat1foralland.Inaddition,(1)impliesthatthesequenceisgraphic(inthesensethatthenecessaryandsucientconditionsforasequencetoberealizedbyagraph[18]aresatis ed)exceptthatwedonotrequirethe'stobeintegers.Wedenotearandomgraphwithagivenexpecteddegreesequence).Forexample,atypicalrandomgraphn;p)(see[19])onverticesandedgedensityisjustarandomgraphwithexpecteddegreesequencepn;pn;:::;pn).Therandomgraph)isdi erentfromtherandomgraphswithanexactdegreesequencesuchasthecon gurationmodel(morediscussioninSection8).In[31,32],MolloyandReedobtainedresultsonthesizesofcon-nectedcomponentsforrandomgraphswithexactdegreesequenceswhichsatisfycertain\smoothing"conditions.Therearealsoanumberofevolutionmodelsforgeneratingapower-lawdegreerandomgraphsasinBollobas,Spenceretal.[11],CooperandFreeze[17]andAiello,ChungandLu[2].InSection8,someofthesemodelswillbediscussed.Herewegivesomede nitions.Theexpectedaveragedegreeofarandom graph)isde nedtobe Forasubsetofvertices,thevolumeof,denotedbyVol(),isthesumofexpecteddegreesinVol(Inparticular,thevolumeVol()of)isjust.Theedgeprobabilityin(1)canbewrittenas: Vol(where Vol( Aconnectedcomponentissaidtobe-smallforan2ifthevolumeofisatmostVol().Wesaythatacomponentis-giantifitsvolumeisatleastVol(),forsomeconstant0.Agiantcomponent,ifexists,isalmostsurelyunique(tobeprovedlaterinSection6).Forasubsetofvertices,atypicalmeasureisthenumberofverticesinthatwecallthesizeof.Intheclassicalrandomgraphn;p),agiantcompo-nentisaconnectedcomponenthavingatleastverticesforsomeconstantOurde nitionofthegiantcomponentinvolvesthevolumeinsteadofthesizeoftheconnectedcomponent.Infact,thede nitionforthegiantcomponentusingthesizeofthecomponentsimplydoesnotworkforrandomgraphswithgeneraldegreedistributions,asillustratedinthefollowingexample.Example1:Weconsiderthatthedegreesequenceconsistingofverticeswithweight2andtheotherverticeswithweight0.Hereisaconstantsatisfy- 1.Therandomgraph)isaunionofarandomgraph andsomeisolatedvertices.Therefore,thelargestconnectedcomponentdoesnothave()vertices.Iftheaveragedegreesatis es,whereisapositiveconstant,wewillshowthatalmostsurelyany-smallconnectedcomponenthassizeat most(log)(detailedinTheorems1-2).Suchcomponentswillbecalledsmallcomponents.AnupperboundforthesizesofsmallcomponentswillbegivenintermsoftheaveragedegreeHerewestatethemainresultswhichwillbeprovedinsubsequentsections.Theorem1Foranypositiveand (1+2,inarandomgraphinwithaveragedegree,almostsurelyeveryconnectedcom-ponenthasvolumeeitheratleastorhassizeatmostlog 1+loglog4+2log(1Theupperboundlog 1+loglog4forsmallcomponentsisasymptoticallybestpossi-bleforlargeTheorem2Foranypositiveandsatisfying 1�2 ,inarandomgraphinwithaveragedegree,everyconnectedcomponentalmostsurelyhasvolumeeitheratleastorhasatmostlog logvertices.Theupperboundlog logisasymptoticallybestpossible.Weconsiderthesecond-orderaveragedegreewhichistheweightedaverageofthesquaresofthevertexweights.Namely,Clearly, PwiPwi Fortheclassicalrandomgraphsn;p),wehave.IntheseminalpaperofErd}osandRenyi[19],itwasshownthatforany0,thereisagiantcomponentif,andthereisnogiantcomponentifFurthermore,adoublejumpoccursnear=1,withthelargestcomponentofsize()if).Forrandomgraphs)ofgeneraldegreedistribution,theevolutionismorecomplicated.Theorem3Forarandomgraphwithagivenexpecteddegreesequencehavingaveragedegree,almostsurelyhasauniquegiantcomponent. (i),thevolumeoftheuniquegiantcomponentisalmostsurelyatleast p (1))Vol((ii),thevolumeoftheuniquegiantcomponentisalmostsurelyatleast1+log (1))Vol(Ifthesecond-orderaveragedegree,thenalmostsurely,thereisnogiantcomponent.TheproofofTheorem3isgiveninSection7.Anaturalquestionarisesconcerningtherelationshipofthedegreestotheemergenceofthegiantcompo-nentfortherangeof.TheexamplesinSection3illustrateboththeexistenceandnon-existenceforsomedegreedistributionssatisfyingNumerousquestionsarise.Forexample,itwouldbeofinteresttocharacterizedegreesequencesforwhichthephasetransitionoccursat=1.Forwhatdegreesequences,aretheredoublejumpsatsuchphasetransition?2BasicfactsandexamplesWewillusethefollowinginequalitywhichisageneralizationoftheCherno inequalitiesforbinomialdistribution:Lemma1Let;:::;Xbeindependentrandomvariableswith=1)=;Pr=0)=1For,wehaveandwede neThenwehaveXE(2)舀XE a=(3)where=max Inequality(3)isacorollaryofageneralconcentrationinequality(seeTheo-rem2.7inthesurveypaperbyMcDiarmid[30]).Inequality(2)whichisaslightimprovementoftheinequalityin[30]canbeprovedasfollows.Proof:Forany01,and0,wede ne+ln(1and2.Thenwehave(0)=(0)=0,and(0)=(0)=0.Also, (1�p+�x)2p�p)e�x (2p (1�p)e�x�(1�p)e�x)2p(1�p)e�x (p Hencewehave+ln(12forany0.Forany0,wehave+(1 2:E(e�t(X�Pniaipi)nYie�t(Xi�piai)nYi=1epi(i)2 2=ePnipi(i)2 2=et2 Wehave 2�t=e�2 bychoosing .ThiscompletestheproofofLemma1. AsimmediateconsequencesofLemma1,thefollowingfactsthenfollow.Fact1:Foragraph),withprobability1,thenumberedgesincidenttoavertexsatis es andProb(1+Fact2:Withprobability1,thenumber)ofedgesin,satis esVol( Vol(Intheotherdirection,Prob(1+)Vol(Vol(Withprobability1 ,allverticessatisfy log log (2 loglogFact3:Withprobabilityatleast1,thenumberofedges)betweenpairsofverticesinisatleast Vol(Vol( Intheremainderofthissection,wewillgiveseveralexampleswithproofswhichillustratethesharpnessofthemainresults.Theseexamplesarealsoinstrumentalfordevelopingmethodslateronfordealingwithrandomgraphswithgivenexpecteddegreedistributions.Example2:Forthefollowingchoicesof1and1,arandomgraphin)almostsurelyhasnogiantcomponent.beaconstantsatisfying10.Foreachofthe rstvertices,theweightissettobe1.Foreachoftheremainingvertices,theweightissettobesatisfying log)and�C�nn: (Forexample,wecanchooselog 2and=10.)WehaveVol(Vol( Vol Vol( �)n�1�+C denotetheinducedgraphonthesetofverticeswithweight1anddenotetheinducedsubgraphon,thesetofverticeswithweightFrom[19],ifNp�,almostsurelyN;p)hasagiantcomponent.IfNp,thenalmostsurelyN;p)doesnothaveagiantcomponentandallcomponentshavesizesatmost(logToapplytheaboveresultto,weselectand Thus,wehave1andconsequentlyalmostsurelyallcomponentsofhavesizeatmost(log(logWewillnextshowthatthereisnogiantcomponentin.We rstconstructanauxiliarygraphfromasfollows.Anewvertexisaddedto,andisconnectedtoallverticesinbuttonovertexin.Thefollowingfactsareimmediate.1.Everyconnectedcomponentofmustbecontainedinsomecomponenthasaspecialcomponentcontainingandallverticesin3.Componentsofotherthanarecomponentsof,whichalmostsurelyhaveatmost(log)verticesandvolume(logNowwewilluseabranchingprocessstartingfromtorevealthecomponent.Forasubset,wede nethe-boundary�u;S.WehaveForeach,theprobabilitythat)ismxmx(1) Thesizeof�()canbeupperboundedbyasumofindependent0-1variables.Theprobabilityofeachrandomvariablehavingvalueoneisaboutmx.Theserandomvariablesaremutuallyindependent.UsingLemmasatis es wherewechoosemxmx:Withprobabilityatleast1(1),thesizeof�()isatmost.Notethat�)=�()iscompletelycontainedin,andsoaretheboundary�)forall2.Sincein,almostsurelyanybranchingprocesscanexpandatmost(log)vertices,thetotalsizeofisalmostsurelyatmostmxO(log+1=logThevolumeofisalmostsurelyatmost2mxO(loglogHenceeachcomponentinalmostsurelycanhavevolumeatmostlog)andconsequentlythereisnogiantcomponentinExample3:Forthefollowingchoiceofdegreesequence1and1,arandomgraphin)almostsurelyhasagiantcomponent.beaverylargebut xedconstant.Foreachofthe rst vertices,theweightissettobe(1).Fortheremaining vertices,eachweightissetto1+.Inthisexample,wehaveVol( Mx+ (1) Vol( (1) Vol Vol(=1+(1)Notethat)containsaclassicalrandomgraphN;p),where and .Since MM+o =1+(1)1,almostN;p)hasagiantcomponentofsize()=().Thecomponentcontainingthisconnectedsubsethasatleast()verticesandatleast(Vol())edges. 3TheexpectednumberofcomponentsofsizeInthissection,weconsidertheprobabilityofhavingaconnectedcomponentofSupposethatwehaveasubsetofverticeswithweights;:::;w.TheprobabilitythatthereisnoedgeleavingS;vVol()(Vol(Vol((4)isaconnectedcomponent,theinducedsubgraphoncontainsatleastonespanningtree.TheprobabilityofcontainingaspanningtreeHencetheprobabilityofhavingaconnectedspanninggraphonisatmostwhererangesoverallspanningtreesonByageneralizedversionofthematrix-treeTheorem[34],theabovesumequalsthedeterminantofany1by1principalsub-matrixofthematrix,whereisthematrixandisthediagonalmatrixdiag(Vol(;:::;w(Vol().Byevaluatingthedeterminant,weconcludethatVol((5)Bycombining(4)and(5),wehaveprovedthefollowing: Lemma2TheexpectedvalueofthenumberofconnectedcomponentsofsizeisatmostVol(Vol(Vol(Vol((6)wherethesumrangesoverallsetsvertices.Lemma3Forapositive,Theexpectedvalueofthenumberof-smallconnectedcomponentsofsizeisatmostVol(Vol((7)wherethesumrangesoverallsetsverticeswithVol(Vol(4ProofofTheorem1Supposethatisarandomgraphin)withexpectedaveragedegree.Wewanttoshowthattheexpectednumber)of-smallcomponentsofsizeissmall.FromLemma3,itsucestoupperboundVol(Vol(Byusingthefactthatthefunctionachievesitsmaximumvalue=(2),wehaveVol(Vol( Vol(Vol( kk(2k�2 1�)2k�2e�(2k�nk k!k�1 kk(2k�2 1�)2k�2e�k�1 4(k�2()k(2 1�)2ke�k1 4(k�2(4 (1�)2)k Theaboveinequalityisusefulwhen whichisanassumptionforTheorem1.Ifsatis eslog 1+loglog(4)2log 1+loglog(4),then 4(k�2=o(1 logsatis es2log 1+loglog(4),wehave 4n2(k�2=o(1 logForlog 1+loglog(4),theprobabilitythatasmallcomponenthassize�kkisatmost�kklog 1+loglog(4) log log(1)Therefore,almostsurelythesizeofa-smallcomponentisatmostlog 1+loglog4.Wehaveprovedthe rstpartofTheorem1.Toshowthattheaboveupperboundisasymptoticallybestpossibleforlarge,weconsiderthefollowingexample.Example4:Weconsiderarandomgraphwiththefollowingweightsastheexpecteddegreesequence.HereweassumethatThereareverticeswithweights(+1.Eachoftheremainingverticeshasweight1.Theaverage(weight)degreeisexactlydenotethesetofverticeswithweight1,anddenotethesetofverticeswithweight(+1.Letbetheinducedgraphof,for2.ThegraphisaclassicalrandomgraphN;p)withand+1))=( ).Almostsurelyisconnected.Infact,iscontainedinthegiantcomponentof.Letdenotethefractionofvertices,whichisnotinthegiantcomponent.Weclaimthatisboundedawayfrom0.Toprovetheclaim,weconsideraspecialbranchingprocess.We rstrevealalledgesin.Thenweexaminetheboundaryof,the2-boundaryof,andsoon,whicheventuallyexposingallverticesinthegiantcomponentof .Foranyvertex,theprobabilityofin�()is n2=31�e�1 Thesizeof�()canbewellapproximatedbythebinomialdistributionwithand .Thuswithhighprobability,thesizeisabout(1 .Wewillestimatethesizeof�)for1byinduction.Supposeishighlyconcentratedonforsomeconstant,for.Foranyvertexnotinnji�i(Si),theprobabilityof)is ain1�e�ai Thesizeof�)canbeapproximatedbythebinomialdistributionwithand .Bythede nitionof.Wehave d):ci=ci�ai+1=cie�ai d:ci=c1iYk=1e�ak =(1 d)e�1�ci Fromtheaboverecurrencefor,weseethatthelimit=limandsatis es=(1 d)e�1�c Itiseasytoseethattheaboveequationhasauniquesolutionofin[01]for1andthesolutionforincreasesasafunctionof.Sincewechooseisboundedawayfromzero.Theclaimisproved.Thesizeofthesecondlargestcomponentcanbeestimatedasfollows.Afterremovingthegiantcomponentfrom,theremaininggraphisaclassicalrandomgrapht;p)withand =c .By[19],thelargestcomponentof )with1hassizeatmostlog2loglog c log (1+(1))log loglog d Theconstant loglog isasymptoticallycloseto 1+loglog4largeandisarbitrarilysmall.ThiscompletestheproofforTheorem1.Remark:Fortheclassicalrandomgraph )with,itwasshown[19]thatthesizeofthesecondlargestconnectedcomponentsisapproximatelythesameasthesizeofthelargestconnectedcomponentof ),wheretheuniquesolutionofforin(01),and .From[19],thelargestcomponentof )hassizeaboutlog2loglog log(1+(1))log logwhichisconsistentwithTheorem1.5ProofofTheorem2Inthissection,weconsider 1�2 .ThemethodsforprovingTheorem1nolongerworkandadi erentestimatefor)isneededhere.Wewillderiveanupperboundfortheexpectednumber)ofconnectedcomponentsofsizebyusinginequality(2).First,wesplit)intotwopartsasfollows:whereVol(kVol(Vol(Vol(Vol(Vol(Tobound),wenotethatisanincreasingfunctionwhen).ThuswehaveVol(Vol( sinceVol(k).ThisimpliesVol(kVol(Vol(Vol(k Vol(Vol(Vol(k kk()2k�2e��)nkk�1 kk()2k�2e��)nk k!k�1 kk()2k�2e��)1 d2k2()kd2ke�(d�)�k=n dk2(d Next,weconsider).Sinceisadecreasingfunctionwhen),wehaveVol(Vol(byusingVol(dk� .Therefore,wehaveVol(Vol(Vol(Vol(Vol( k!k�1()k�2e��)1 d2k2dke�(d(1�)�kn dk2(d Togetherwehave 2(de ed�)�1)k: log log2log log,wehave 2=O(1 log2log log,then 2=O(1 logBysettinglog log,theprobabilityofhavingasmallcomponentof�kkisatmost�kklog log log log(1)Therefore,almostsurelythesizeofasmallcomponentisatmostlog logToseethatthisupperboundisbestpossible,weconsiderthefollowingexample.Intherandomgraph )with1,thelargestcomponenthassizeaboutlog2loglog log(see[19]),asdesired.6ProofofTheorem3BeforeprovingTheorem3,we rstproveseveralreductions.Fact4:Supposethatarandomgraph)hasaveragedegreeandcontainsaconnectedsubsethavingmorethanlogvertices,where log.ThenalmostsurelythereisagiantcomponentinProof:Fact4isanimmediateconsequenceofTheorems1and2,subjecttoverifyingtherequiredassumptionswhichfollowfromthede nitionoffollows: log logforsome0when1+d2,and 1+loglog4+2log(1forsome0when2Fact5:Aninducedsubgraph)isarandomgraphwithgivenexpectedsequencewhichconsistsofVol(for TheprooffollowsfromthefactthattheexpecteddegreeofisjustVol(Lemma4Supposethatinarandomgraph,thereisavalue(independentof)sothatforall,andtheaverageexpecteddegree,whereisapositiveconstant.Thenalmostsurelyhasauniquegiantcomponent.Proof:Weuseabranchingprocessasfollows:Firstchooseanyvertexweightgreaterthan1andcarryoutabreadth rstsearchofitsconnectedcomponent.Avertexiscalledunexaminedifithasbeendiscoveredtobeinthecomponent,butwehavenotyetexposeditsneighbors.Letbethesumoftheweightsofallunexaminedverticesatdepth.SupposeVol()and.Foranyvertexnotyetbeenexposed,theprobabilitythatistobediscoveredinthecomponentatdepth+1is.Hence,theexpectedvalueofunexposedByLemma1,wehavewhereBychoosing2,wehave 2(~d�)Xk)e�(~d�1�)2 Foreachincreasesbyafactorof 1withthefailureprobabilityatmost,where converges,thereexistsaconstantsatisfyingforapositiveconstant.Withapositiveconstantprobability,willincreaseatleastbyafactorof +1)isanabsoluteconstant,theeventoccurswithsomepositiveconstantprobability.Ifthebranchingprocessdiesearly(i.e.,theconnectedcomponentissmall),thenwejuststartanotherbranchingprocessfromanewvertexwithweightgreaterthan1.(Thereareenoughsuchverticessincethenumberofverticeswithweightgreaterthan1isatleast .)Afteratmost(log)tries,almostsurelythegiantcomponentwillberevealed. Theproofoftheorem3:1beaconstantsatisfying(1(forexample,choose4).Wesorttheverticessothat.LetdenotethelargestintegersatisfyingVol( ,weuseFact5whichimpliesthereisaninducedsubgraphonverticeshavingexpecteddegrees2.ItcontainstheErd}os-Renyigraph))andthereforeitcontainsacomponentof)verticesforsomeconstant ,weconsidertheinducedsubgraphonthe rstvertices.ByFact5,ithasvolume(1andthereforehasaveragedegreeatleast1+Furthermore,allweightsareboundedbytheconstant2.ByLemma4,itcontainsacomponentofvolumeFrom(1),maximumweightisnomorethan Vol().Bothandhaveatleast vertices.ByFact4,thegiantcomponentalmostsurelyexists.Theaboveargumentscanbeusedtoshowtheuniquenessofthegiantcom-ponentaswell.Foranytwoverticesand,webeginabranchingprocessstartingatbutstopatthemomentwhenthevolumeofthesetofexposedverticesreaches (2+)Vol()log.Thenwebeginanewbranchingprocessstartingatandstopatthemomentwhenthevolumeoftheexposedverticesreaches (2+)Vol()log.ThentheprobabilityofandbeingnotconnectedbyanyedgeisatmostVol()Vol()log Theprobabilitythateverypairofverticeseachinagiantcomponentarecon-nectedisatleast1.Thus,thegiantcomponentisalmostsurelyunique.Next,weconsiderthevolumeofthegiantcomponent.Wewanttoshowthefollowing:(i),thevolumeofthegiantcomponentisatleast(1 p (1))Vol((ii)If1+,thevolumeofthegiantcomponentisatleast1+log (1))Vol(We rstconsiderthecaseof.If(i)doesnothold,thenthegiantcomponentis-smallforsomesatisfying p .ByTheorem1,thesizeofthegiantcomponentisatmostlog 1+loglog4+2log(1.Hencethereisonevertexwithweightgreatthanorequaltotheaverage:Vol( log 1+loglog4+2log(1Vol( logItiseasytocheckthatwhichcontradictsourassumption(1).Hencethevolumeofthegiantcomponentisatleast(1 p (1))Vol()ifForthecaseof,weagainprovebycontradiction.Supposethatallconnectedcomponentis-smallforsomesatisfying1+log .ByTheorem2,thesizeofthegiantcomponentisatmostlog log.Hencethereisonevertexwithweightgreatthanorequaltotheaverage:Vol( log logVol( logThiscontradictstheassumption(1)and(ii)isproved.Itremainstoshowthatforsmallerthan1,almostsurelyallcomponentshavevolumesatmost logandthereforethereisnogiantcomponentinthiscase.Claim:If,withprobabilityatleast1 ,allcomponentshavevolumeatmost n. Proof:Letbetheprobabilitythatthereisacomponenthavingvolumegreaterthan .Nowwechoosetworandomverticeswiththeprobabilityofbeingchosenproportionaltotheirweights.Undertheconditionthatthereisacom-ponentwithvolumegreaterthan ,theprobabilityofeachvertexinthiscomponentisatleast .Therefore,theprobabilitythattherandompairofverticesareinthesamecomponentisatleast xn(8)Ontheotherhand,fora xedpairofverticesand,theprobabilityu;vandbeingconnectedbyapathoflength+1isatmostu;v:::iTheprobabilitythatandbelongtothesamecomponentisatmostu;v Sincetheprobabilitiesofandbeingselectedareandrespectively,theprobabilitythattherandompairofverticesareinthesameconnectedcomponentisatmostu;v 1�~dwuwv=~d2 Combiningwith(8),wehavexn whichimplies Thereforewithprobabilityatmost1 ,allcomponentshavesizeatmost asdesired.Thiscompletestheprooffortheclaim.Bychoosingbelog,wehaveshownthatwithprobabilityatleast1(1),allcomponentsaresmall.WehavecompletedtheproofforTheorem3. 7SeveralrandomgraphmodelsIntheliterature,thefollowingmodel,socalledthecon gurationmodel,isoftenusedtoconstructarandomgraphwithaprescribeddegreesequence.Itwas rstintroducedbyBenderandCan eld[9],re nedbyBollobas[10]andalsoWormald[35].Arandomgraphwithgivendegreesisassociatedwitharandommatchinginasetnodes.Eachvertexcorrespondstoasetnodesin.Thenumberofedgesbetweentwoverticesandisthenumberofedgesintheassociatedmatchingwithonenodeinandonenode.Itiseasytoseethattheresultinggraph(asamulti-graph)hasdegreesexactlyasrequired.MolloyandReed[31,32]usedthecon gurationmodeltoshowthatifthereverticesofdegree,where=1and0,thenthegraphalmostsurelyhasagiantcomponentifthefollowingconditionsaresatis ed.1.Themaximumdegreeisatmosttendsuniformlyto3.Thelimit)=limexists,andthesumapproachesthelimituniformly.4.Thedegreesequenceisgraphic.Theadvantageofthecon gurationmodelistogenerategraphsexactlywiththeprescribeddegreesanditistheprimarymodelforexaminingregulargraphswithconstantdegrees.Thereareseveraldisadvantagesofthecon gurationmodel.Theanalysisofthecon gurationmodelismuchmorecomplicatedduetothedependencyoftheedges.Arandomgraphfromthecon gurationmodelisinfactamultigraphinsteadofasimplegraph.Theprobabilityofhavingmultipleedgesincreasesrapidlywhenthedegreesincrease.InthepapersofMolloyandReed,theconditiononmaximumdegreewithanupperboundofisrequiredbecauseoftheoccurenceofmultipleedgesinthecon gurationmodel.Consequently,thismodelisrestrictiveforpower-lawgraphs,where thelargestdegreecanbequitelarge.Furthermore,additionalconditions(e.g.,Condition2and3asin[31,32])areoftenrequiredforthecon gurationmodels.Inthesameway,theclassicalrandomgraphmodeln;p)isoftenpreferredtothecon gurationmodelsofrandomgraphswithTheadvantageofthegeneralizedmodelthatweusehereisthesimplicitywithoutanyconditiononthedegreesequenceexceptfortheonlyassumption(1).Ourmodeldoesnotproducethegraphwithexactgivendegreesequence.Instead,ityieldsarandomgraphwithgivenexpecteddegreesequence.Anotherlineofapproachwhichsimulatesrealisticgraphsistogenerateavertex/edgeatatime,startingfromonenodeorasmallgraph.Althoughwewillnotdealwithsuchmodelsinthispaper,wewillbrie ymentionseveralevolutionmodels.BarabasiandAlbert[7]describethefollowinggraphevolutionprocess.Startingwithasmallinitialgraph,ateachtimesteptheyaddanewnodeandanedgebetweenthenewnodeandeachofrandomnodesintheexistinggraph,whereisaparameterofthemodel.Therandomnodesarenotchosenuniformly.Instead,theprobabilityofpickinganodeisweightedaccordingtoitsexistingdegree(theedgesareassumedtobeundirected).Usingheuristicanalysiswiththeassumptionthatthediscretedegreedistributionisdi erentiable,theyderiveapowerlawforthedegreedistributionwithapowerof3,regardlessof.Apowerlawwithpower3forthedegreedistributionofthismodelwasindependentlyderivedandprovedbyBollobasetal.[11].Kumaratel.[28]proposedthreeevolutionmodels|\lineargrowthcopy-ing",\exponentialgrowthcopying",and\lineargrowthvariants".TheLineargrowthcopingmodeladdsonenewvertexwithout-linksatatime.Thedesti-nationof-thout-linkofthenewvertexiseithercopiedfromthecorrespondingout-linkofa\prototype"vertex(chosenrandomly)orarandomvertex.Theyshowedthatthein-degreesequencefollowsthepowerlaw.ThesemodelsweredesignedexplicitlytomodeltheWorldWideWeb.Indeed,theyshowthattheirmodelhasalargenumberofcompletebipartitesubgraphs,ashasbeenobservedintheWWWgraph,whereasseveralothermodelsdonot.This(andthelin-eargrowthvariantsmodel)hasthesimilardrawbackasthe rstmodelin[27]. Theout-degreeofeveryvertexisalwaysaconstant.Edgesandverticesintheexponentialgrowthcopyingmodelincreaseexponentially.Aielloetal.describedageneralrandomgraphevolutionprocessin[3]forgeneratingdirectedpowerlawgraphswithgivenexpectedin-degreesandout-degrees.Ateachtime,anewnodeisgeneratedandcertainedgesareaddedasfollows.Theendpointsofnewedgescanbeeitherthenewnodeoroneoftheexistingnodes.Anexistingnodeisselectedasthedestination(ortheorigin)withprobabilityproportionaltoitsin-degree(orout-degree).Therearefourtypesofedgesaccordingtotheirdestinationsandorigins.Aprobabilityspacecontrolsthenumberandthetypeofedgestobeaddedattime.Undertheassumptionthatthenumberofedgesaddedateachtimeisboundedandhasalimitingdistribution,Aielloetal.[3]provedthisgeneralprocessgeneratespowerlawgraphs.Thepowerofthepowerlawofout-degree(orin-degree)equalsto2+ ,whereistheexpectednumberofedgesperstepwiththenewnodeastheorigin(orthedestination)andistheexpectednumberofedgesperstepwithanexistingnodeastheorigin(orthedestination).Recently,CooperandFrieze[17]independentlyanalyzedtheaboveevolutionofaddingeithernewverticesornewedgesandderivedpowerlawdegreedistributionforverticesofsmalldegrees.8RemarksonpowerlawgraphsInthispaper,weexaminethesizesofconnectedcomponentsofarandomgraphwithgivendegreesequences.Theresultsandmethodsherecanbeusefultoexaminepowerlawgraphsthatariseinvariouscontext.Apowerlawgraphwithpowerhasthenumberofverticesofdegreeproportionalto.Forexample,thecollaborationbasedonthedatafromMathematicsReview[22]hasabout337,000vertices(asauthors)andabout496,000edges(asjointpublications).So,theaveragedegreeis2.94.Tomodelthecollaborationgraphasarandompowerlawgraph,theexponentisapproximately97asshowninFigure1.WealsoincludetheactualdataonthesizesofconnectedcomponentsinFigure 100 1000 10000 100000 1 10 100 1000 the number of vertices degree 10 100 1000 10000 100000 1 10 100 1000 10000 100000 1e+06 the number of connected components size Figure1:Degreedistributionofthecol-laborationgraph.Figure2:Connectedcomponentdistri-bution.[1]L.A.AdamicandB.A.Huberman,GrowthdynamicsoftheWorldWideWeb,Nature,September9,1999,pp.131.[2]W.Aiello,F.ChungandL.Lu,Arandomgraphmodelformassivegraphs,ProceedingsoftheThirty-SecondAnnualACMSymposiumonTheoryofComputing,(2000)171-180.[3]W.Aiello,F.ChungandL.Lu,Randomevolutioninmassivegraphs,ExtendedabstractappearedinThe42thAnnualSymposiumonFounda-tionofComputerSciences,October,2001.PaperversionhasappearedinHandbookonMassiveDataSets,(Eds.J.Abelloetal.),KluwerAcademicPublishers,(2002),97-122."[4]L.A.N.Amaral,A.Scala,M.Barthelemy,andH.E.Stanley,Classesofsmall-worldnetworks,Proc.Natl.Acad.Sci.USA,vol.,no.21,(2000),[5]N.AlonandJ.H.Spencer,TheProbabilisticMethod,WileyandSons,NewYork,1992. [6]R.B.R.AzevedoandA.M.Leroi,Apowerlawforcells,Proc.Natl.Acad.Sci.USA,vol.,no.10,(2001),5699-5704.[7]Albert-LaszloBarabasiandRekaAlbert,Emergenceofscalinginrandomnetworks,Science(1999)509-512.[8]A.Barabasi,R.Albert,andH.Jeong,Scale-freecharacteristicsofrandomnetworks:thetopologyoftheworldwideweb,PhysicaA272(1999),173-[9]E.A.BenderandE.R.Can eld,Theasymptoticnumberoflabelledgraphswithgivendegreesequences,J.Combinat.Theory(A),(1978),296-307.[10]B.BollobRandomGraphs,Academic,NewYork,1985.[11]B.Bollobas,O.Riordan,J.SpencerandG.Tusnady,TheDegreeSe-quenceofaScale-FreeRandomGraphProcess,RandomStructuresandAlgorithms,Vol.,no.3(2001),279-290.[12]A.Broder,R.Kumar,F.Maghoul,P.Raghavan,S.Rajagopalan,R.Stata,A.Tompkins,andJ.Wiener,\GraphStructureintheWeb,"proceedingsoftheWWW9Conference,May,2000,Amsterdam.PaperversionappearedComputerNetworks,(1-6),(2000),309-321.[13]K.Calvert,M.Doar,andE.Zegura,ModelingInternettopology.CommunicationsMagazine(1997)160-163.[14]FanChungandLinyuanLu,Thediameterofrandomsparsegraphs,Ad-vancesinAppliedMath.(2001),257-279.[15]F.ChungandL.Lu,Averagedistancesinrandomgraphswithgivenex-pecteddegreesequences,Proc.Natl.Acad.Sci.,toappear.[16]F.Chung,L.LuandV.Vu,Eigenvaluesofrandompowerlawgraphs,preprint. [17]C.CooperandA.Frieze,Ageneralmodelofwebgraphs,ProceedingsofESA,500-511."[18]P.Erd}osandT.Gallai,Grafokel}irtfokupontokkal(Graphswithpointsofprescribeddegrees,inHungarian),Mat.Lapok(1961),264-274.[19]P.Erd}osandA.Renyi,Onrandomgraphs.I,Publ.Math.Debrecen(1959),290-291.[20]M.Faloutsos,P.Faloutsos,andC.Faloutsos,Onpower-lawrelationshipsoftheInternettopology,ProceedingsSIGCOMM,1999,251-262.[21]N.Gilbert,Asimulationofthestructureofacademicscience,SocialogicalResearchOnline(2),1997.[22]J.Grossman,TheErd}osNumberProject,http://www.oakland.edu/grossman/erdoshp.html.[23]S.JainandS.Krishna,Amodelfortheemergenceofcooperation,interde-pendence,andstructureinevolvingnetworks,Proc.Natl.Acad.Sci.USAvol.,no.2,(2001),543-547.[24]S.Janson,T. Luczak,andA.RucRandomGraphs,Wiley-Interscience,NewYork,2000.[25]J.Kleinberg,S.R.Kumar,P.Raghavan,S.RajagopalanandA.Tomkins,Thewebasagraph:Measurements,modelsandmethods,ProceedingsoftheInternationalConferenceonCombinatoricsandComputing,1999,26-[26]R.Kumar,P.Raghavan,S.RajagopalanandA.Tomkins,Trawlingthewebforemergingcybercommunities,Proceedingsofthe8thWorldWideWebConference,Toronto,1999.ExtendedversioninComputerNetworksVol.31,No.11-16,(1999),1481-1493. 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