Scale-free random graphs and Potts model

《Scale-free random graphs and Potts model》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、PRAMANA°cIndianAcademyofSciencesVol.64,No.6

2、journalofJune2005physicspp.1149{1159Scale-freerandomgraphsandPottsmodelD-SLEE1;2,K-IGOH1,BKAHNG1andDKIM11SchoolofPhysicsandCenterforTheoreticalPhysics,SeoulNationalUniversity,Seoul151-747,Korea2TheoretischePhysik,Univ

3、ersitÄatdesSaarlandes,66041SaarbrÄucken,GermanyAbstract.Weintroduceasimplealgorithmthatconstructsscale-freerandomgraphse±ciently:eachvertexihasaprescribedweightP/i¡¹(0<¹<1)andanedgeicanconnectverticesiandjwithratePiPj.Correspondingequilibriumensembleisidenti¯ed

4、andtheproblemissolvedbytheq!1limitoftheq-statePottsmodelwithinhomogeneousinteractionsforallpairsofspins.Thenumberofloopsaswellasthegiantclustersizeandthemeanclustersizeareobtainedinthethermodynamiclimitasafunctionoftheedgedensity.Variouscriticalexponentsassocia

5、tedwiththepercolationtransitionarealsoobtainedtogetherwith¯nite-sizescalingforms.Theprocessofformingthegiantclusterisqualitativelydi®erentbetweenthecasesof¸>3and2<¸<3,where¸=1+¹¡1isthedegreedistributionexponent.Whilefortheformer,thegiantclusterformsabruptlyatth

6、epercolationtransition,forthelatter,however,theformationofthegiantclusterisgradualandthemeanclustersizefor¯niteNshowsdoublepeaks.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