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1、ReducingDegeneracyinMaximumEntropyModelsofNetworksSzabolcsHorvát,1évaCzabarka,2andZoltánToroczkai11DepartmentofPhysics,UniversityofNotreDame,NotreDame,IN,46556USA2DepartmentofMathematics,UniversityofSouthCarolina,Columbia,SC,29208USABasedonJaynes’smaximumentropyprinciple,exponenti
2、alrandomgraphsprovideafamilyofprincipledmodelsthatallowthepredictionofnetworkpropertiesasconstrainedbyempiricaldata.However,theiruseisoftenhinderedbythedegeneracyproblemcharacterizedbyspontaneoussymmetry-breaking,wherepredictionssimplyfail.Hereweshowthatdegeneracyappearswhenthecor
3、respondingdensityofstatesfunctionisnotlog-concave.Weproposeasolutiontothedegeneracyproblemforalargeclassofmodelsbyexploitingthenonlinearrelationshipsbetweentheconstrainedmeasurestoconvexifythedomainofthedensityofstates.Wedemonstratethee?ectivenessofthemethodonexamples,includingonZ
4、achary’skarateclubnetworkdata.PACSnumbers:89.75.Hc,89.70.Cf,05.20.-y,87.23.GeOurunderstandingandmodelingofcomplexsystemspresentouranalysisandresultsusingthelanguageofisalwaysbasedonpartialinformation,limiteddataandnetworksandERGmodels,however,our?ndingsareknowledge.Theonlyprincipl
5、edmethodofpredictinggenerallyapplicable.LetusconsiderthesetGNofallpropertiesofacomplexsystemsubjecttowhatisknownlabeledsimplegraphs(noparalleledges,orself-loops)on(dataandknowledge)isbasedontheMaximumEntropyNnodes,representingthemicrostates7!G,andanPrincipleofJaynes[1,2].Usingthi
6、sprinciple,here-arbitrarysetofgraphmeasures,orobservablesm(G)=derivedtheformalismofstatisticalmechanics,bothclas-m1(G);:::;mK(G),e.g.,thenumberofedgesmj,2-starssical[1]andthetime-dependentquantumdensity-matrixm_,trianglesmM,thedegreeofthe9thnode.Theseformalism[2],usingShannon’sinf
7、ormationentropy[3].measuresrepresenttheconstraintsandweassumethatThemethodgeneratesaprobabilitydistributionP()wearegivenspeci?cvaluesm0,forthem(inputdata).overallthepossible(micro)statesofthesystembyTheymaycomefromanempiricalnetworkG0,orcouldPmaximizingtheentropyS[P]=
