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1、PRAMANA°cIndianAcademyofSciencesVol.64,No.6
2、journalofJune2005physicspp.1149{1159Scale-freerandomgraphsandPottsmodelD-SLEE1;2,K-IGOH1,BKAHNG1andDKIM11SchoolofPhysicsandCenterforTheoreticalPhysics,SeoulNationalUniversity,Seoul151-747,Korea2TheoretischePhysik,Univ
3、ersit?atdesSaarlandes,66041Saarbr?ucken,GermanyAbstract.Weintroduceasimplealgorithmthatconstructsscale-freerandomgraphse±ciently:eachvertexihasaprescribedweightP/i?1(0<1<1)andanedgeicanconnectverticesiandjwithratePiPj.Correspondingequilibriumensembleisidentiˉed
4、andtheproblemissolvedbytheq!1limitoftheq-statePottsmodelwithinhomogeneousinteractionsforallpairsofspins.Thenumberofloopsaswellasthegiantclustersizeandthemeanclustersizeareobtainedinthethermodynamiclimitasafunctionoftheedgedensity.Variouscriticalexponentsassocia
5、tedwiththepercolationtransitionarealsoobtainedtogetherwithˉnite-sizescalingforms.Theprocessofformingthegiantclusterisqualitativelydi?erentbetweenthecasesof?>3and2
6、epercolationtransition,forthelatter,however,theformationofthegiantclusterisgradualandthemeanclustersizeforˉniteNshowsdoublepeaks.Keywords.Scale-freerandomgraph;percolationtransition;Pottsmodel.PACSNos89.70.+c;89.75.-k;05.70.Jk1.IntroductionGraphtheoreticapproac
7、hisofgreatvaluetocharacterizecomplexsystemsfoundinsocial,informationalandbiologicalareas.Here,asystemisrepresentedasagraphornetworkwhoseverticesandedgesstandforitsconstituentsandinteractions.AsimplemodelforsuchnetworksistherandomgraphmodelproposedbyErd}osandR?e
8、nyi(ER)[1].IntheERmodel,Nnumberofverticesarepresentfromthebeginningandedgesareaddedonebyoneinthesystem,connectingpairsofverticesselectedrandomly.ThedegreedistributionisPoissonian.However,manyreal-worldnetworkssuchastheworld-wideweb,theInternet,thecoauthorship,t
9、heproteininteractionnetworksandsoondisplaypower-lawbehaviorsinthedegreedistribution.Suchnetworksarecalledscale-free(SF)networks[2].ThankstorecentextensivestudiesofSFnetworks
