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1、CorrelatedrandomnetworksJohannesBergandMichaelL¨assigInstitutf¨urTheoretischePhysik,Universit¨atzuK¨olnZ¨ulpicherStraße77,50937K¨oln,Germany(February1,2008)Wedevelopastatisticaltheoryofnetworks.Anetworkisasetofverticesandlinksgivenbyitsadjacencymatri
2、xc,andtherelevantstatisticalensemblesaredefinedintermsofapartitionfunctionPZ=exp[−βH(c)].Thesimplestcasesareuncorrelatedrandomnetworkssuchasthewell-knowncErd¨os-R´enygraphs.HerewestudymoregeneralinteractionsH(c)whichleadtocorrelations,forexample,betwe
3、entheconnectivitiesofadjacentvertices.Inparticular,suchcorrelationsoccurinoptimizednetworksdescribedbypartitionfunctionsinthelimitβ→∞.Theyarearguedtobeacrucialsignatureofevolutionarydesigninbiologicalnetworks.PACSnumbers:89.75.Hc89.75.-k05.20.yNetwor
4、ksdescribestructuresasdiverseastheinter-ensemblesofrelevanceherehaveafiniteaveragecon-actionlinksbetweenproteinsinacell,thewiringofthenectivityκ≡2M/N.Forfixedκ,thedistributionofbrain,ortheconnectionsoftheinternet.Recenttheoret-connectivities,icalwork[1
5、,2]hasfocusedoncommunicationnetworks,XNandawealthofquantitativedataisnowbecomingavail-1p(k)≡hδ(ki−k)i,(2)ableonnetworksinmolecularbiology.ExamplesincludeNi=1controlnetworksingenetranscription[3],theinteractionmapofproteins[4],andthepathwaysofcellmeta
6、bolismbecomesasymptoticallyindependentofN,implyingthat[5].AllthesesystemsconsistofmanydifferentkindsoftypicaladjacencymatricescbecomesparseforlargeN.moleculeslinkedbycomplexinteractions.Networkmod-Thisisthecaseofinterestforapplications.elsareasimplifie
7、ddescription,whichneglectsquantita-Asatisfactorymathematicaltheoryexiststodateonlytiveaspectsoftheseinteractionsandfocusessolelyonforwhatwecalluncorrelatedrandomnetworks[8,9].Intheirpathways.thiscase,theHamiltoniandependsonlyonsingle-pointAnetworkisa
8、setofverticesi=1,...,Nconnectedconnectivities,bylinks.ItisuniquelydefinedbytheadjacencymatrixXNc,whoseentriesarecij=1ifthereisalinkfromitojH1(c)=f(ki),(3)andcij=0otherwise.Weconsiderherenetworkswithi=1undirectedlinks,wherecissymmetric.Theconnectivityo
