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1、PHYSICALREVIEWE74,0361212006Exactsolutionsformodelsofevolvingnetworkswithadditionanddeletionofnodes1,2,34,53,4CristopherMoore,GourabGhoshal,andM.E.J.Newman1DepartmentofComputerScienceandDepartmentofPhysicsandAstronomy,UniversityofNewMexico,Albuquerque,NewMexico87131,USA2
2、SantaFeInstitute,SantaFe,NewMexico87501,USA3CenterfortheStudyofComplexSystems,UniversityofMichigan,AnnArbor,Michigan48109,USA4DepartmentofPhysics,UniversityofMichigan,AnnArbor,Michigan48109,USA5MichiganCenterforTheoreticalPhysics,UniversityofMichigan,AnnArbor,Michigan,4810
3、9,USAReceived11April2006;published28September2006Therehasbeenconsiderablerecentinterestinthepropertiesofnetworks,suchascitationnetworksandtheworldwideweb,thatgrowbytheadditionofvertices,andanumberofsimplesolvablemodelsofnetworkgrowthhavebeenstudied.Intherealworld,however
4、,manynetworks,includingtheweb,notonlyaddverticesbutalsolosethem.Hereweformulatemodelsofthetimeevolutionofsuchnetworksandgiveexactsolutionsforanumberofcasesofparticularinterest.Forthecaseofnetgrowthandso-calledpreferentialattachmentinwhichnewlyappearingverticesattachtoprevi
5、ouslyexistingonesinproportiontovertexdegreeweshowthattheresultingnetworkshavepower-lawdegreedistributions,butwithanexponentthatdivergesasthegrowthratevanishes.Weconjecturethatthelowexponentvaluesobservedinreal-worldnetworksarethustheresultofvigorousgrowthinwhichtherateofad
6、ditionofverticesfarexceedstherateofremoval.Weregrowthtoslowinthefutureforinstance,inamorematurefutureversionofthewebwewouldexpecttoseeexponentsincrease,potentiallywithoutbound.DOI:10.1103/PhysRevE.74.036121PACSnumbers:89.75.Hc,89.20.Hh,05.10.Gg,05.65.bI.INTRODUCTIONdele
7、tionaffectsthecrucialpower-lawbehaviorofthedegreedistribution,whileinothercasesitdoesnot.ThestudyofnetworkshasattractedasubstantialamountInthispaper,westudythegeneralprocessinwhichaofattentionfromthephysicscommunityinthelastfewyearsnetworkgrowsor,potentially,shrinksbythe
8、constantad-13,inpartbecauseofnetworksbroadutilityasrepresen-ditionandremovalofverticesa
