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1、PHYSICALREVIEWE70,056131(2004)AnalysisofweightednetworksM.E.J.NewmanDepartmentofPhysicsandCenterfortheStudyofComplexSystems,UniversityofMichigan,AnnArbor,Michigan48109-1120,USAandSantaFeInstitute,1399HydeParkRoad,SantaFe,NewMexico87501,USA(Received26July2004;published24November2004)Theconnections
2、inmanynetworksarenotmerelybinaryentities,eitherpresentornot,buthaveassociatedweightsthatrecordtheirstrengthsrelativetooneanother.Recentstudiesofnetworkshave,byandlarge,steeredclearofsuchweightednetworks,whichareoftenperceivedasbeinghardertoanalyzethantheirunweightedcounterparts.Herewepointoutthat
3、weightednetworkscaninmanycasesbeanalyzedusingasimplemappingfromaweightednetworktoanunweightedmultigraph,allowingustoapplystandardtech-niquesforunweightedgraphstoweightedonesaswell.Wegiveanumberofexamplesofthemethod,includinganalgorithmfordetectingcommunitystructureinweightednetworksandasimpleproo
4、fofthemaximum-¯ow±minimum-cuttheorem.DOI:10.1103/PhysRevE.70.056131PACSnumber(s):89.75.Hc,05.10.2a,87.23.GeI.INTRODUCTIONgraphs,wepointout,manyofthestandardtechniquesthathavebeendevelopedtostudyunweightednetworkscanbeManysystemscanusefullyberepresentedasnetworksorcarriedoverwithlittleornomodi®cat
5、iontotheweightedgraphsÐcollectionsofverticesjoinedinpairsbyedges.Ex-case.Wepresentanumberofexamplesofapplicationstoamplesincludetheinternetandtheworldwideweb,citationwellknownnetworkproblems.networks,socialnetworks,andbiologicalandbiochemicalnetworksofvariouskinds.Althoughanoldandwellestab-lished
6、branchofstudyinmathematicsandsociology,re-II.WEIGHTEDNETWORKSANDMULTIGRAPHSsearchonnetworkshasinrecentyearsattractedsigni®cantattentionfrommembersofthephysicscommunityaswell,whohavesuccessfullyappliedavarietyofphysicalideastoAweightednetworkcanberepresentedmathematicallytheanalysisandmodelingofth
7、esesystems[1±4].byanadjacencymatrixwithentriesthatarenotsimplyzeroMostofthenetworksthathavebeenstudiedinthephys-or1,butareequalinsteadtotheweightsontheedges:icsliteraturehavebeenbinaryinnature;thatis,theedgesbetweenver
