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1、Fractalityincomplexnetworks:criticalandsupercriticalskeletonsJ.S.Kim,1K.-I.Goh,2G.Salvi,1E.Oh,1B.Kahng,1,3andD.Kim11SchoolofPhysicsandAstronomyandCenterforTheoreticalPhysics,SeoulNationalUniversity,Seoul151-747,Korea2CenterforCancerSystemsBiology,Dana-FarberCancerInstitute,HarvardMedicalScho
2、ol,Boston,MA02115andCenterforComplexNetworkResearchandDepartmentofPhysics,UniversityofNotreDame,NotreDame,IN465563CenterforNonlinearStudies,LosAlamosNationalLaboratory,LosAlamos,NM87545(Dated:February6,2008)Fractalscaling—apower-lawbehaviorofthenumberofboxesneededtotileagivennetworkwithrespe
3、cttothelateralsizeofthebox—isstudied.Weintroduceanewbox-coveringalgorithmthatisamodifiedversionoftheoriginalalgorithmintroducedbySongetal.[Nature(London)433,392(2005)];thisalgorithmenableseffectivecomputationandeasyimplementation.Fractalnetworksareviewedascomprisingaskeletonandshortcuts.Theske
4、leton,embeddedunderneaththeoriginalnetwork,isaspecialtypeofspanningtreebasedontheedgebetweennesscentrality;itprovidesascaffoldforthefractalityofthenetwork.Whentheskeletonisregardedasabranchingtree,itexhibitsaplateauinthemeanbranchingnumberasafunctionofthedistancefromaroot.Fornon-fractalnetwor
5、ks,ontheotherhand,themeanbranchingnumberdecaystozerowithoutformingaplateau.Basedontheseobservations,weconstructafractalnetworkmodelbycombiningarandombranchingtreeandlocalshortcuts.Thescaffoldbranchingtreecanbeeithercriticalorsupercritical,dependingonthesmall-worldnessofagivennetwork.Forthenet
6、workconstructedfromthecritical(supercritical)branchingtree,theaveragenumberofverticeswithinagivenboxgrowswiththelateralsizeoftheboxaccordingtoapower-law(anexponential)forminthecluster-growingmethod.Thecriticalandsupercriticalskeletonsareobservedinproteininteractionnetworksandtheworld-wideweb
7、,respectively.Thedistributionofboxmasses,i.e.,thenumberofverticeswithineach−ηbox,followsapowerlawPm(M)∼M.TheexponentηdependsontheboxlateralsizeℓB.ForsmallvaluesofℓB,ηisequaltothedegreeexponentγofagivenscale-freenetwork,whereasηapproachestheexponent
