Generation of uncorrelated random scale-free networks

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1、Generationofuncorrelatedrandomscale-freenetworksMicheleCatanzaro,1Mari´anBogu˜n´a,2andRomualdoPastor-Satorras11DepartamentdeF´ısicaiEnginyeriaNuclear,UniversitatPolit`ecnicadeCatalunya,CampusNordB4,08034Barcelona,Spain2DepartamentdeF´ısicaFonamental,Universit

2、atdeBarcelona,Av.Diagonal647,08028Barcelona,Spain(Dated:February2,2008)Uncorrelatedrandomscale-freenetworksareusefulnullmodelstochecktheaccuracyantheanalyticalsolutionsofdynamicalprocessesdefinedoncomplexnetworks.Weproposeandanalyzeamodelcapabletogeneraterando

3、muncorrelatedscale-freenetworkswithnomultipleandself-connections.Themodelisbasedontheclassicalconfigurationmodel,withanadditionalrestrictiononthemaximumpossibledegreeofthevertices.Wechecknumericallythattheproposedmodelindeedgeneratesscale-freenetworkswithnotwo

4、andthreevertexcorrelations,asmeasuredbytheaveragedegreeofthenearestneighborsandtheclusteringcoefficientoftheverticesofdegreek,respectively.PACSnumbers:89.75.-k,87.23.Ge,05.70.LnComplexnetworksconstituteageneralframeworkforvertexofdegreekisconnectedtoavertexofde

5、greek′.thetopologicalcharacterizationofmanynaturalandFromanumericalpointofview,itismoreconvenienttotechnologicalsystemswhosecomplexitypreventsamorecharacterizedegree-degreecorrelationsbymeansofthedetailedmicroscopicdescription[1,2,3].Withinthisaveragedegreeof

6、thenearestneighborsoftheverticesofframework,thesesystemsarerepresentedintermsofdegreek,whichisformallydefinedas[10]networksorgraphs[4],inwhichverticesstandfortheXk¯(k)=k′P(k′

7、k).(2)unitscomposingthesystem,whileedgesamongverticesnnrepresenttheinteractionsorrela

8、tionsbetweenpairsofk′units.Thefocusisthusshifttothetopologicalcharac-Degree-degreecorrelationshasledtoafirstclassifica-terizationoftherepresentativenetwork,ataskwhichistionofcomplexnetworksaccordingtothisproperty[11].largelymorefeasibleandyields,nevertheless,an

9、otice-Thus,whenk¯nn(k)isanincreasingfunctionofk,thecor-ableamountofinformationonthestructureandprop-respondingnetworkissaidtoexhibitassortativemixingertiesoftheoriginalsystem.Theempirical