cut-offs and finite size effects in scale-free networks

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1、EPJmanuscriptNo.(willbeinsertedbytheeditor)Cut-offsandfinitesizeeffectsinscale-freenetworksMari´anBogu˜n´a1,RomualdoPastor-Satorras2,andAlessandroVespignani31DepartamentdeF´ısicaFonamental,UniversitatdeBarcelona,AvingudaDiagonal647,08028Barcelona,Spain2DepartamentdeF´ısicaiEnginyeriaNuclear,Univers

2、itatPolit`ecnicadeCatalunya,CampusNord,08034Barcelona,Spain3LaboratoiredePhysiqueTh´eorique(UMR8627duCNRS),Bˆatiment210,Universit´edeParis-Sud,91405OrsayCedex,FranceReceived:date/Revisedversion:dateAbstract.Weanalyzethedegreedistribution’scut-offinfinitesizescale-freenetworks.Weshowthatthecut-offbe

3、haviorwiththenumberofverticesNisruledbythetopologicalconstraintsinducedbytheconnectivitystructureofthenetwork.Eveninthesimplecaseofuncorrelatednetworks,weobtainanexpressionofthestructuralcut-offthatissmallerthatthenaturalcut-offobtainedbymeansofextremaltheoryarguments.Theobtainedresultsareexplicit

4、lyappliedinthecaseoftheconfigurationmodeltorecoverthesizescalingoftadpolesandmultipleedges.PACS.89.75.-k–87.23.Ge–05.70.Ln1IntroductiontoestimatethecriticalpropertiesofdynamicalsystemsdefinedinnetworkswithSFtopologies.ThepresenceofboundedSFdistributionsincomplexRecentyearshavewitnessedanincreasing

5、scientificinter-networkshasbeenobservedinseveralsystems[14].Inestforthestudyofcomplexnetworksandthedynami-somecases,theboundordegreecut-offcanbeexplainedcalprocessestakingplaceontopofthem[1,2].Indeed,intermsofafinitecapacityoftheverticestocollectcon-thecomplextopologicalpropertiesshownbymanyrealnec

6、tionsorduetoincompleteinformation[14,15].Inthisnetworkshavelargeeffectsonthebehaviorofseveralphe-case,thevalueofthecut-offisaconstantthatdependsnomenacharacterizingthedynamicsandstabilityoftheseonthephysicalconstraintsactingonthesystems.Asec-systems.Theseeffectsareparticularlyintriguingintheondposs

7、ibility,theoneinwhichweareinterestedhere,caseofscale-free(SF)networks,thatis,innetworksintakesplacewhenthecut-offispurelyaccountedforbythewhichtheprobabilityP(k)thatavertexisconnectedtokfinitesizeofthenetwork,asusuallyhappensi