Vertex similarity in networks

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1、VertexsimilarityinnetworksE.A.Leicht,PetterHolme,andM.E.J.NewmanDepartmentofPhysics,UniversityofMichigan,AnnArbor,MI48109,U.S.A.Weconsidermethodsforquantifyingthesimilarityofverticesinnetworks.Weproposeameasureofsimilaritybasedontheconceptthattwoverticesaresimilariftheiri

2、mmediateneighborsinthenetworkarethemselvessimilar.Thisleadstoaself-consistentmatrixformulationofsimilaritythatcanbeevaluatediterativelyusingonlyaknowledgeoftheadjacencymatrixofthenetwork.Wetestoursimilaritymeasureoncomputer-generatednetworksforwhichtheexpectedresultsarekn

3、own,andonanumberofreal-worldnetworks.PACSnumbers:89.75.Fb,89.75.HcI.INTRODUCTIONANDBACKGROUNDneighbors.Forinstance,itmaybereasonabletoconcludethattwoindividualsinasocialnetworkhavesomethingincommonThestudyofnetworkedsystems,includingcomputernet-iftheysharemanyofthesamefri

4、ends.Letibetheneighbor-works,socialnetworks,biologicalnetworks,andothers,hasat-hoodofvertexiinanetwork,i.e.,thesetofverticesthataretractedconsiderableattentionintherecentphysicsliterature(Al-directlyconnectedtoiviaanedge.Thenthenumberofcom-bertandBarabási,2002;Dorogovtse

5、vandMendes,2002;New-monfriendsofiandjisman,2003).Anumberofstructuralpropertiesofnetworkshavebeenthesubjectofparticularlyintensescrutiny,includ-unnorm=jijj,(1)ingthelengthsofpathsbetweenvertices(WattsandStrogatz,1998;ChungandLu,2002;CohenandHavlin,2003;Fron-wherejxjind

6、icatesthecardinality(i.e.,numberofelementsin)czaketal.,2004),degreedistributions(BarabásiandAlbert,thesetx,sothatjij,forinstance,issimplyequaltothedegree1999;Dorogovtsevetal.,2000;Krapivskyetal.,2000),com-ofvertexi.munitystructure(GirvanandNewman,2002;Radicchietal.,Thequ

7、antityunnormcanberegardedasarudimentarymea-2004;Guimeràetal.,2004;DonettiandMuñoz,2004),andsureofsimilaritybetweeniandj.Itis,however,notentirelyvariousmeasuresofvertexcentrality(WattsandStrogatz,1998;satisfactory.ItcantakelargevaluesforverticeswithhighdegreeNewman,2001;G

8、ohetal.,2001;Holmeetal.,2002).evenifonlyasmallfractionoftheirneighborsarethesame,andAnotherimpor