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1、CliquepercolationinrandomnetworksImreDer´enyi,1GergelyPalla,2andTam´asVicsek1,21DepartmentofBiologicalPhysics,E¨otv¨osUniversity,P´azm´anyP.stny.1A,H-1117Budapest,Hungary2BiologicalPhysicsResearchGroupofHAS,P´azm´anyP.stny.1A,H-1117Budapest,Hungary(s
2、ubmittedtoPhys.Rev.Lett.10November2004;acceptedforpublication20April2005)Thenotionofk-cliquepercolationinrandomgraphsisintroduced,wherekisthesizeofthecompletesubgraphswhoselargescaleorganizationsareanalyticallyandnumericallyinvestigated.FortheErd˝os-
3、R´enyigraphofNverticesweobtainthatthepercolationtransitionofk-cliquestakesplacewhentheprobabilityof−1/(k−1)twoverticesbeingconnectedbyanedgereachesthethresholdpc(k)=[(k−1)N].AtthetransitionpointthescalingofthegiantcomponentwithNishighlynon-trivialand
4、dependsonk.Wediscusswhycliquepercolationisanovelandefficientapproachtotheidentificationofoverlappingcommunitiesinlargerealnetworks.PACSnumbers:02.10.Ox,89.75.Hc,05.70.Fh,64.60.-iTherehasbeenaquicklygrowinginterestinnetworks,percolationintheERmodel.Weob
5、tainanalyticandsimu-sincetheycanrepresentthestructureofawideclassofcom-lationresultsrelatedtotheappearanceofagiantcomponentplexsystemsoccurringfromthelevelofcellstosociety.Datamadeofcompletesubgraphsofkvertices(k-cliques).Inpar-obtainedonrealnetworks
6、showthatthecorrespondinggraphsticular,weprovideananalyticexpressionforthethresholdexhibitunexpectednon-trivialproperties,e.g.,anomalousde-probabilityatwhichthepercolationtransitionofk-cliquesgreedistributions,diameter,spreadingphenomena,cluster-takes
7、place.Thetransitioniscontinuous,characterizedbyingcoefficient,andcorrelations[1,2,3,4,5].Veryre-non-universalcriticalexponents,whichdependonbothkandcentlygreatattentionhasbeenpaidtothelocalstructuralthewaythesizeofthegiantcomponentismeasured.Ourana-un
8、itsofnetworks.Smallandwelldefinedsubgraphshavelyticcalculationsareinfullagreementwiththecorrespondingbeenintroducedas“motifs”[6].Theirdistributionandclus-numericalsimulations.teringproperties[6,7,8]canbeusedtointerpretglobalBeforeweproceedtocalculatet
