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1、PHYSICALREVIEWE70,066117(2004)StatisticalmechanicsofnetworksJuyongParkandM.E.J.NewmanDepartmentofPhysicsandCenterfortheStudyofComplexSystems,UniversityofMichigan,AnnArbor,Michigan48109-1120,USA(Received2June2004;revisedmanuscriptreceived20August2004;published7December2004)Westudyt
2、hefamilyofnetworkmodelsderivedbyrequiringtheexpectedpropertiesofagraphensembletomatchagivensetofmeasurementsofareal-worldnetwork,whilemaximizingtheentropyoftheensemble.ModelsofthistypeplaythesameroleinthestudyofnetworksasisplayedbytheBoltzmanndistributioninclassicalstatisticalmech
3、anics;theyofferthebestpredictionofnetworkpropertiessubjecttotheconstraintsimposedbyagivensetofobservations.Wegiveexactsolutionsofmodelswithinthisclassthatincorporatearbitrarydegreedistributionsandarbitrarybutindependentedgeprobabilities.Wealsodiscusssomemorecomplexexampleswithcorr
4、elatededgesthatcanbesolvedapproximatelyorexactlybyadaptingvariousfamiliarmethods,includingmean-®eldtheory,perturbationtheory,andsaddle-pointexpansions.DOI:10.1103/PhysRevE.70.066117PACSnumber(s):89.75.Hc,05.20.2y,87.23.Ge,89.20.HhI.INTRODUCTIONtheoryrapidlybecomescomplexanddif®cul
5、ttouseifweattempttomakeitrealisticbytheinclusionofaccuratein-Thelastfewyearshaveseenthepublicationofalargetermolecularpotentialsandsimilarfeatures.Inpractice,ki-volumeofworkinthephysicsliteratureonnetworksofvari-netictheorymodelseithermakeonlyratherroughandun-ouskinds,particularly
6、computerandinformationnetworkscontrolledpredictions,orrelyonlarge-scalecomputerliketheinternetandworldwideweb,biologicalnetworkssimulationtoachieveaccuracy.suchasfoodwebsandmetabolicnetworks,andsocialnet-Ifonewantsagoodcalculationaltoolforstudyingtheworks[1±4].Thisworkhasbeendivid
7、edbetweenempiricalpropertiesofgases,therefore,onedoesnotusekineticstudiesofthestructureofparticularnetworksandtheoreticaltheory.Instead,oneusesstatisticalmechanics.Althoughcer-studiesfocusedlargelyonthecreationofmathematicalandtainlylessintuitive,statisticalmechanicsisbasedonrigor
8、-computationalmodels.Theconstruct
