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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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
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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]aresatised)exceptthatwedonotrequirethe'stobeintegers.Wedenotearandomgraphwithagivenexpecteddegreesequence).Forexample,atypicalrandomgraphn;p)(see[19])onverticesandedgedensityisjustarandomgraphwithexpecteddegreesequencepn;pn;:::;pn).Therandomgraph)isdierentfromtherandomgraphswithanexactdegreesequencesuchasthecongurationmodel(morediscussioninSection8).In[31,32],MolloyandReedobtainedresultsonthesizesofcon-nectedcomponentsforrandomgraphswithexactdegreesequenceswhichsatisfycertain\smoothing"conditions.Therearealsoanumberofevolutionmodelsforgeneratingapower-lawdegreerandomgraphsasinBollobas,Spenceretal.[11],CooperandFreeze[17]andAiello,ChungandLu[2].InSection8,someofthesemodelswillbediscussed.Herewegivesomedenitions.Theexpectedaveragedegreeofarandom graph)isdenedtobe 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-nentisaconnectedcomponenthavingatleastverticesforsomeconstantOurdenitionofthegiantcomponentinvolvesthevolumeinsteadofthesizeoftheconnectedcomponent.Infact,thedenitionforthegiantcomponentusingthesizeofthecomponentsimplydoesnotworkforrandomgraphswithgeneraldegreedistributions,asillustratedinthefollowingexample.Example1:Weconsiderthatthedegreesequenceconsistingofverticeswithweight2andtheotherverticeswithweight0.Hereisaconstantsatisfy- 1.Therandomgraph)isaunionofarandomgraph andsomeisolatedvertices.Therefore,thelargestconnectedcomponentdoesnothave()vertices.Iftheaveragedegreesatises,whereisapositiveconstant,wewillshowthatalmostsurelyany-smallconnectedcomponenthassizeat most(log)(detailedinTheorems1-2).Suchcomponentswillbecalledsmallcomponents.AnupperboundforthesizesofsmallcomponentswillbegivenintermsoftheaveragedegreeHerewestatethemainresultswhichwillbeprovedinsubsequentsections.Theorem1Foranypositiveand (1+2,inarandomgraphinwithaveragedegree,almostsurelyeveryconnectedcom-ponenthasvolumeeitheratleastorhassizeatmostlog 1+loglog4+2log(1Theupperboundlog 1+loglog4forsmallcomponentsisasymptoticallybestpossi-bleforlargeTheorem2Foranypositiveandsatisfying 12 ,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,wehaveandwedeneThenwehaveXE(2)舀XE a=(3)where=max Inequality(3)isacorollaryofageneralconcentrationinequality(seeTheo-rem2.7inthesurveypaperbyMcDiarmid[30]).Inequality(2)whichisaslightimprovementoftheinequalityin[30]canbeprovedasfollows.Proof:Forany01,and0,wedene+ln(1and2.Thenwehave(0)=(0)=0,and(0)=(0)=0.Also, (1p+x)2pp)ex (2p (1p)ex(1p)ex)2p(1p)ex (p Hencewehave+ln(12forany0.Forany0,wehave+(1 2:E(et(XPniaipi)nYiet(Xipiai)nYi=1epi(i)2 2=ePnipi(i)2 2=et2 Wehave 2t=e2 bychoosing .ThiscompletestheproofofLemma1. AsimmediateconsequencesofLemma1,thefollowingfactsthenfollow.Fact1:Foragraph),withprobability1,thenumberedgesincidenttoavertexsatises andProb(1+Fact2:Withprobability1,thenumber)ofedgesin,satisesVol( 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.Foreachoftherstvertices,theweightissettobe1.Foreachoftheremainingvertices,theweightissettobesatisfying log)andCnn: (Forexample,wecanchooselog 2and=10.)WehaveVol(Vol( Vol Vol( )n1+C denotetheinducedgraphonthesetofverticeswithweight1anddenotetheinducedsubgraphon,thesetofverticeswithweightFrom[19],ifNp,almostsurelyN;p)hasagiantcomponent.IfNp,thenalmostsurelyN;p)doesnothaveagiantcomponentandallcomponentshavesizesatmost(logToapplytheaboveresultto,weselectand Thus,wehave1andconsequentlyalmostsurelyallcomponentsofhavesizeatmost(log(logWewillnextshowthatthereisnogiantcomponentin.Werstconstructanauxiliarygraphfromasfollows.Anewvertexisaddedto,andisconnectedtoallverticesinbuttonovertexin.Thefollowingfactsareimmediate.1.Everyconnectedcomponentofmustbecontainedinsomecomponenthasaspecialcomponentcontainingandallverticesin3.Componentsofotherthanarecomponentsof,whichalmostsurelyhaveatmost(log)verticesandvolume(logNowwewilluseabranchingprocessstartingfromtorevealthecomponent.Forasubset,wedenethe-boundaryu;S.WehaveForeach,theprobabilitythat)ismxmx(1) Thesizeof()canbeupperboundedbyasumofindependent0-1variables.Theprobabilityofeachrandomvariablehavingvalueoneisaboutmx.Theserandomvariablesaremutuallyindependent.UsingLemmasatises wherewechoosemxmx:Withprobabilityatleast1(1),thesizeof()isatmost.Notethat)=()iscompletelycontainedin,andsoaretheboundary)forall2.Sincein,almostsurelyanybranchingprocesscanexpandatmost(log)vertices,thetotalsizeofisalmostsurelyatmostmxO(log+1=logThevolumeofisalmostsurelyatmost2mxO(loglogHenceeachcomponentinalmostsurelycanhavevolumeatmostlog)andconsequentlythereisnogiantcomponentinExample3:Forthefollowingchoiceofdegreesequence1and1,arandomgraphin)almostsurelyhasagiantcomponent.beaverylargebutxedconstant.Foreachoftherst 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(2k2 1)2k2e(2knk k!k1 kk(2k2 1)2k2ek1 4(k2()k(2 1)2kek1 4(k2(4 (1)2)k Theaboveinequalityisusefulwhen whichisanassumptionforTheorem1.Ifsatiseslog 1+loglog(4)2log 1+loglog(4),then 4(k2=o(1 logsatises2log 1+loglog(4),wehave 4n2(k2=o(1 logForlog 1+loglog(4),theprobabilitythatasmallcomponenthassizekkisatmostkklog 1+loglog(4) log log(1)Therefore,almostsurelythesizeofa-smallcomponentisatmostlog 1+loglog4.WehaveprovedtherstpartofTheorem1.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.Werstrevealalledgesin.Thenweexaminetheboundaryof,the2-boundaryof,andsoon,whicheventuallyexposingallverticesinthegiantcomponentof .Foranyvertex,theprobabilityofin()is n2=31e1 Thesizeof()canbewellapproximatedbythebinomialdistributionwithand .Thuswithhighprobability,thesizeisabout(1 .Wewillestimatethesizeof)for1byinduction.Supposeishighlyconcentratedonforsomeconstant,for.Foranyvertexnotinnjii(Si),theprobabilityof)is ain1eai Thesizeof)canbeapproximatedbythebinomialdistributionwithand .Bythedenitionof.Wehave d):ci=ciai+1=cieai d:ci=c1iYk=1eak =(1 d)e1ci Fromtheaboverecurrencefor,weseethatthelimit=limandsatises=(1 d)e1c 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 12 .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()2k2e)nkk1 kk()2k2e)nk k!k1 kk()2k2e)1 d2k2()kd2ke(d)k=n dk2(d Next,weconsider).Sinceisadecreasingfunctionwhen),wehaveVol(Vol(byusingVol(dk .Therefore,wehaveVol(Vol(Vol(Vol(Vol( k!k1()k2e)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,theprobabilityofhavingasmallcomponentofkkisatmostkklog log log log(1)Therefore,almostsurelythesizeofasmallcomponentisatmostlog logToseethatthisupperboundisbestpossible,weconsiderthefollowingexample.Intherandomgraph )with1,thelargestcomponenthassizeaboutlog2loglog log(see[19]),asdesired.6ProofofTheorem3BeforeprovingTheorem3,werstproveseveralreductions.Fact4:Supposethatarandomgraph)hasaveragedegreeandcontainsaconnectedsubsethavingmorethanlogvertices,where log.ThenalmostsurelythereisagiantcomponentinProof:Fact4isanimmediateconsequenceofTheorems1and2,subjecttoverifyingtherequiredassumptionswhichfollowfromthedenitionoffollows: log logforsome0when1+d2,and 1+loglog4+2log(1forsome0when2Fact5:Aninducedsubgraph)isarandomgraphwithgivenexpectedsequencewhichconsistsofVol(for TheprooffollowsfromthefactthattheexpecteddegreeofisjustVol(Lemma4Supposethatinarandomgraph,thereisavalue(independentof)sothatforall,andtheaverageexpecteddegree,whereisapositiveconstant.Thenalmostsurelyhasauniquegiantcomponent.Proof:Weuseabranchingprocessasfollows:Firstchooseanyvertexweightgreaterthan1andcarryoutabreadthrstsearchofitsconnectedcomponent.Avertexiscalledunexaminedifithasbeendiscoveredtobeinthecomponent,butwehavenotyetexposeditsneighbors.Letbethesumoftheweightsofallunexaminedverticesatdepth.SupposeVol()and.Foranyvertexnotyetbeenexposed,theprobabilitythatistobediscoveredinthecomponentatdepth+1is.Hence,theexpectedvalueofunexposedByLemma1,wehavewhereBychoosing2,wehave 2(~d)Xk)e(~d1)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 ,weconsidertheinducedsubgraphontherstvertices.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(Werstconsiderthecaseof.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,foraxedpairofverticesand,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,socalledthecongurationmodel,isoftenusedtoconstructarandomgraphwithaprescribeddegreesequence.ItwasrstintroducedbyBenderandCaneld[9],renedbyBollobas[10]andalsoWormald[35].Arandomgraphwithgivendegreesisassociatedwitharandommatchinginasetnodes.Eachvertexcorrespondstoasetnodesin.Thenumberofedgesbetweentwoverticesandisthenumberofedgesintheassociatedmatchingwithonenodeinandonenode.Itiseasytoseethattheresultinggraph(asamulti-graph)hasdegreesexactlyasrequired.MolloyandReed[31,32]usedthecongurationmodeltoshowthatifthereverticesofdegree,where=1and0,thenthegraphalmostsurelyhasagiantcomponentifthefollowingconditionsaresatised.1.Themaximumdegreeisatmosttendsuniformlyto3.Thelimit)=limexists,andthesumapproachesthelimituniformly.4.Thedegreesequenceisgraphic.Theadvantageofthecongurationmodelistogenerategraphsexactlywiththeprescribeddegreesanditistheprimarymodelforexaminingregulargraphswithconstantdegrees.Thereareseveraldisadvantagesofthecongurationmodel.Theanalysisofthecongurationmodelismuchmorecomplicatedduetothedependencyoftheedges.Arandomgraphfromthecongurationmodelisinfactamultigraphinsteadofasimplegraph.Theprobabilityofhavingmultipleedgesincreasesrapidlywhenthedegreesincrease.InthepapersofMolloyandReed,theconditiononmaximumdegreewithanupperboundofisrequiredbecauseoftheoccurenceofmultipleedgesinthecongurationmodel.Consequently,thismodelisrestrictiveforpower-lawgraphs,where thelargestdegreecanbequitelarge.Furthermore,additionalconditions(e.g.,Condition2and3asin[31,32])areoftenrequiredforthecongurationmodels.Inthesameway,theclassicalrandomgraphmodeln;p)isoftenpreferredtothecongurationmodelsofrandomgraphswithTheadvantageofthegeneralizedmodelthatweusehereisthesimplicitywithoutanyconditiononthedegreesequenceexceptfortheonlyassumption(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)hasthesimilardrawbackastherstmodelin[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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