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

Presentation on theme: "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"— Presentation transcript

ConnectedcomponentsinrandomgraphswithgivenexpecteddegreesequencesFanChungLinyuanLuAbstractWeconsiderafamilyofrandomgraphswithagivenexpecteddegreesequence.Eachedgeischosenindependentlywithprobabilitypropor-tionaltotheproductoftheexpecteddegreesofitsendpoints.Weexaminethedistributionofthesizes/volumesoftheconnectedcomponentswhichturnsoutdependingprimarilyontheaveragedegreeandthesecond-orderaveragedegree.Heredenotestheweightedaverageofsquaresoftheexpecteddegrees.Forexample,weprovethatthegiantcomponentexistsiftheexpectedaveragedegreeisatleast1,andthereisnogiantcomponentiftheexpectedsecond-orderaveragedegreeisatmost1.Examplesaregiventoillustratethatbothboundsarebestpossible.1IntroductionTheprimarysubjectinthestudyofrandomgraphtheoryistheclassicalrandomgraphn;p),asintroducedbyErd}osandRenyiin1959[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(inthesensethatthenecessaryandsucientconditionsforasequencetoberealizedbyagraph[18]aresatised)exceptthatwedonotrequirethe'stobeintegers.Wedenotearandomgraphwithagivenexpecteddegreesequence).Forexample,atypicalrandomgraphn;p)(see[19])onverticesandedgedensityisjustarandomgraphwithexpecteddegreesequencepn;pn;:::;pn).Therandomgraph)isdierentfromtherandomgraphswithanexactdegreesequencesuchasthecongurationmodel(morediscussioninSection8).In[31,32],MolloyandReedobtainedresultsonthesizesofcon-nectedcomponentsforrandomgraphswithexactdegreesequenceswhichsatisfycertain\smoothing"conditions.Therearealsoanumberofevolutionmodelsforgeneratingapower-lawdegreerandomgraphsasinBollobas,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, PwiPwi Fortheclassicalrandomgraphsn;p),wehave.IntheseminalpaperofErd}osandRenyi[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?2BasicfactsandexamplesWewillusethefollowinginequalitywhichisageneralizationoftheChernoinequalitiesforbinomialdistribution: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)2p�p)e�x (2p (1�p)e�x�(1�p)e�x)2p(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)ismxmx(1) Thesizeof�()canbeupperboundedbyasumofindependent0-1variables.Theprobabilityofeachrandomvariablehavingvalueoneisaboutmx.Theserandomvariablesaremutuallyindependent.UsingLemmasatises wherewechoosemxmx: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,itsucestoupperboundVol(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�k1 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=31�e�1 Thesizeof�()canbewellapproximatedbythebinomialdistributionwithand .Thuswithhighprobability,thesizeisabout(1 .Wewillestimatethesizeof�)for1byinduction.Supposeishighlyconcentratedonforsomeconstant,for.Foranyvertexnotinnji�i(Si),theprobabilityof)is ain1�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 .ThemethodsforprovingTheorem1nolongerworkandadierentestimatefor)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��)nkk�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 d2k2dke�(d(1�)�kn 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-Renyigraph))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nedbyBollobas[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.BarabasiandAlbert[7]describethefollowinggraphevolutionprocess.Startingwithasmallinitialgraph,ateachtimesteptheyaddanewnodeandanedgebetweenthenewnodeandeachofrandomnodesintheexistinggraph,whereisaparameterofthemodel.Therandomnodesarenotchosenuniformly.Instead,theprobabilityofpickinganodeisweightedaccordingtoitsexistingdegree(theedgesareassumedtobeundirected).Usingheuristicanalysiswiththeassumptionthatthediscretedegreedistributionisdierentiable,theyderiveapowerlawforthedegreedistributionwithapowerof3,regardlessof.Apowerlawwithpower3forthedegreedistributionofthismodelwasindependentlyderivedandprovedbyBollobasetal.[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.Barthelemy,andH.E.Stanley,Classesofsmall-worldnetworks,Proc.Natl.Acad.Sci.USA,vol.,no.21,(2000),[5]N.AlonandJ.H.Spencer,TheProbabilisticMethod,WileyandSons,NewYork,1992. 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