Percolation transition in correlated static model

arXiv:0802.2644v1 [cond-mat.stat-mech] 19 Feb 2008PercolationTransitioninCorrelatedStaticModel

Sang-WooKimandJaeDongNoh∗

DepartmentofPhysics,UniversityofSeoul,Seoul130-743,Korea(Received13September2007)

Weintroduceacorrelatedstaticmodelandinvestigateapercolationtransition.Themodelisamodificationofthestaticmodelandischaracterizedbyassortativedegree-degreecorrelation.Asonevariestheedgedensity,thenetworkundergoesapercolationtransition.Thepercolationtransitionischaracterizedbyaweaksingularbehaviorofthemeanclustersizeandpower-lawscalingsofthepercolationorderparameterandtheclustersizedistributionintheentirenon-percolatingphase.Theseresultssuggestthattheassortativedegree-degreecorrelationgeneratesaglobalstructuralcorrelationwhichisrelevanttothepercolationcriticalphenomenaofcomplexnetworks.

PACSnumbers:89.75.Hc,05.10.-a,05.70.Fh,05.50.+qKeywords:Percolation,Degree-degreecorrelation,Powerlaw,Criticalphenomena

I.INTRODUCTION

Manysystemsinnaturehaveacomplexnetworkstruc-

ture.Forexample,theInternetisawirednetworkof

computersandrouters,theWorldWideWebisanetwork

ofwebpageshyperlinkedtoeachother,aproteininter-

actionnetworkisanetworkofinteractingproteinsinan

organism,andasocialnetworkisanetworkofindividuals

whoarelinkedthroughacertainrelationship.Suchcom-

plexnetworkshaveahighlyinhomogeneousstructure.In

ordertocharacterizeandunderstandthestructureand

dynamics,extensiveresearchhasbeenperformedforthe

lastdecade[1,2,3,4,5,6,7].

Adegreedistributionp(k)fortheprobabilityofanode

havingdegreekisaquantityofprimaryimportance.Itis

oneoftheessentialcharacteristicsthatinfluencesstruc-

turalproperties,dynamicalbehaviors,andcollectivephe-

nomenaofcomplexnetworks.However,thedegreeisa

propertyofeachindividualvertex.Hence,thedegree

distributionbyitselfcannotdescribecorrelationamong

differentvertices.Itisattractinggrowinginterest,since

manyreal-worldnetworksdisplayacertainlevelofcor-

relation,whichhasasignificantimpactonnetworkprop-

erties[8,9,10,11].

Thedegree-degree(DD)correlationreferstothecor-

relationbetweendegreesofneighboringvertices[8].The

wholeinformationonthecorrelationiscontainedinthe

degreecorrelationfunctionp(k′,k),theprobabilityofan

edgelinkingnodesofdegreekandk′,andthecon-

ditionalprobabilityp(k′|k)≡p(k′,k)/(󰀁

k′′p(k′′,k)),

theprobabilityofanodeamongneighborsofdegree-k

nodeshavingdegreek′.Theoverallfeatureisconve-

nientlycharacterizedbytheassortativityandthemean

neighbordegreefunction.Theassortativitycoefficientr

isdefinedasthenormalizedPearsoncorrelationcoeffi-

cientbetweendegreesofneighboringvertices[8],andthe

meanneighbordegreefunctionisdefinedasKNN(k)≡󰀁

k′k′p(k′|k)[12,13].Anetworkwithapositive(neg-2

toasthecorrelatedstaticmodel.Basicpropertiesofthemodelarealsopresented.InSec.III,weinvestigatebond

percolationtransitionsinthecorrelatedstaticmodel.We

summarizeandconcludethepaperinSec.IV.

II.CORRELATEDSTATICMODEL

Thestaticmodelisanefficientmodelforuncorre-

latednetworks[20].Astatic-modelnetworkwithNver-

ticesandKedgesisconstructedasfollows:(i)Each

vertexi(i=1,···,N)isassignedtoaweightwi=

i−µ/(󰀁Nj=1j−µ);(ii)accordingtotheprobability{wi},

twoverticesarechosenatrandom.Ifthereisnoedgebe-

tweenthem,theyarelinkedwithanedge.Theprocedure

(ii)isrepeateduntilonehasKedgesintotal.Aresulting

networkisscale-freewithapower-lawdegreedistribution

p(k)∼k−λwiththedegreeexponentλ=1+1/µ.When

µ=0,allverticesarechosenwithequalprobabilityand

themodelreducestotheErd˝os-R´enyirandomnetwork

withthePoissondegreedistribution.Thestatic-model

networkisuncorrelatedforµ<1/2orλ>3[21].The

percolationtransitioninthismodelhasbeenthoroughly

studied[22].

Wemodifythestaticmodeltoincorporatetheassor-

tativeDDcorrelation.AnetworkwithNverticesand

Kedgesinthecorrelatedstaticmodelisconstructedas

follows:(i)Eachvertexi(i=1,···,N)isassignedto

aweightwi=i−µ/(󰀁Nj=1j−µ);(ii)avertexiischo-

senwithprobabilitywiatrandom,andthenanother

vertexjischosenrandomlyamongallverticeswiththe

samedegreeasi.Ifthereisnoedgebetweeniandj,

anedgeconnectingthemisadded.Theprocedure(ii)

isrepeateduntilthereareKedgesintotal.Notethat

thestaticmodelandthecorrelatedstaticmodeldifferin

theprocedure(ii).Whileedgesareaddedbetweenver-

ticeschosenindependentlyintheformer,theyareadded

betweenverticesofthesamedegreeinthelatter.This

generatesapositiveDDcorrelation.

Considerthedegreedistributionofthecorrelated

staticmodel.Letpi(k,t)betheprobabilitythata

vertexihasdegreekattimestept.Sinceanedge

isaddedateachtimestep,thetotalnumberofedges

Kisequaltot.Thedegreedistributionisgivenby

p(k,t)=1

Npk−1(t)󰀄

pi(k−1,t)

−󰀃

wi+qk(t)N[qk−1(t)−qk(t)].(1)

Notethatthecorrelatedstaticmodelhasthesameevo-

lutionequationasthestaticmodel.Henceweconclude

thatthecorrelatedstaticmodelhasthesamedegreedis-

tributionasthestaticmodel.Fromthepropertiesofthe