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DynamicalWeights and Enhanced Synchronization in Adaptive Complex Networks

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DynamicalWeightsandEnhancedSynchronizationinAdaptiveComplexNetworks

¨rgenKurthsChangsongZhouandJu

InstituteofPhysics,UniversityofPotsdamPF601553,14415Potsdam,Germany

(Received7July2005;revisedmanuscriptreceived15November2005;published24April2006)Dynamicalorganizationofconnectionweightsisstudiedinscale-freenetworksofchaoticoscillators,wherethecouplingstrengthofanodefromitsneighborsdevelopsadaptivelyaccordingtothelocalsynchronizationpropertybetweenthenodeanditsneighbors.We ndthatwhencompletesynchroniza-tionisachieved,thecouplingstrengthbecomesweightedandcorrelatedwiththetopologyduetoahierarchicaltransitiontosynchronizationinheterogeneousnetworks.Importantly,suchanadaptiveprocessenhancessigni cantlythesynchronizabilityofthenetworks,whichcouldhavemeaningfulimplicationsinthemanipulationofdynamicalnetworks.

DOI:10.1103/PhysRevLett.96.164102

PACSnumbers:05.45.Xt,87.18.Sn,89.75.ÿk

Real-worldcomplexnetworksareinteractingdynamicalentitieswithaninterplaybetweendynamicalstatesandinteractionpatterns.Whiletopologicalstudieshavere-vealedimportantorganizationprinciplesinthestructures[1],amorecompleteunderstandingwouldrequirecharac-terizationsbeyondthetopology.Therearerecentlyseveralapproachesinthisdirection.Forexample,(i)thestudyofmorerealisticweightedpropertiesoftheconnections[2,3].Theanalysisofsomerealnetworkshasshownthatthecon-nectionweightsareoftenhighlyheterogeneousandcor-´si-Albert(BA)relatedwiththedegrees[2].TheBaraba

model[4]hasbeengeneralizedtotaketheconnectionweightsintoaccount[3].(ii)Intensiveinvestigationsofsynchronizationdynamicsofoscillatorynetworks[5–8].However,mostoftheseworksconsidernetworksthatdonotchangewiththedynamics,andwecallsuchnetworkscrystallizednetworks(CNs)[9].(iii)Thegrowingattentiononuni edstudiesofthecoevolutionofdynamicalstatesandnetworkstructures[10–14].Modelsofadaptivenet-works(ANs)havebeenproposed,e.g.,evolvingofnodesdueto tnessininteractingspecies[11],reinforcementofconnectionstrength[12]orrewiringoflinks[13]duetopayoffsamongagentsplayinggames;oradaptivechangesofcouplingstrengthaccordingtothestatedistanceingloballycoupledchaoticmaps[14]inadesynchronizedregime.

Signi cantrecentinterestinsynchronizationisrelatedtotheidenti cationofthenetworkstructuresthatenhanceoroptimizetheglobalsynchronizability.Theoptimal(un-weighted)con gurationsobtainedbyoptimizationalgo-rithmsareentanglednetworkswithratheruniformdegrees[15],asconsistentwithgraph-theoreticalpredications[7,16].However,suchtopologiesdonot ttomostrealisticnetworksystems.Manycomplexnetworks,wheresyn-chronizationisrelevant,areoftenheterogeneousintopol-ogyandarenaturallyweighted,suchasnetworksofcorticalareas[17],networksofcitiesinthesynchroniza-tionofepidemicoutbreaks[18]andspreading[19].Thusmoreinterestingisthein uenceofweightedpropertiesonthesynchronizabilityindegreeheterogeneousnetworks[7,8,16],sincesuitablyweightedconnectionscanenhancesigni cantlythesynchronizabilitywithoutchangingtheheterogeneityinthetopology[7,16].Aquestionofsub-stantialimportanceinthelightofANsiswhethersuchweightedpropertiesforenhancedsynchronizabilitycanbeself-organized.

InthisLetter,westudyadaptiveweightorganizationandfocusontheimpactsofheterogeneoustopologyontheweightstructureandontheenhancementofglobalsyn-chronizability.Weintroduceasimple,butgeneric,schemeofweightadaptationaccordingtoalocalsynchronizationproperty,whichleadstoglobalsynchronizationofthewholenetwork.Wemainlyshowthat(i)thenetworkbecomesweightedandtheweightsarenegativelycorre-latedwiththedegrees,and(ii)importantly,theadaptationenhancessigni cantlythesynchronizabilitycomparedtounweightedCNs.

WeconsiderNcoupledidenticalchaoticoscillators

_i F xi x

NXj 1

Gij H xj ÿH xi ;

(1)

whereF x isthedynamicsofindividualoscillatorsand

H x isthelinearoutputfunction.G Gij istheweightedcouplingmatrix,Gij AijWij,whereAijisthebinaryadjacencymatrixandWij>0isthecouplingstrengthfromnodejtonodeiiftheyareconnected.Inun-weightednetworks,Wij 1isuniformforalllinks.Inpre-0

viousstudiesofCNs[5–8],Gij G0ij,whereGijis xedand controlstheoverallstrengthoftheconnections.HerewestudyANswherethecouplingstrengthsWijiscontrolledbythelocalsynchronizationpropertiesofthenodes.Toachieveaglobalsynchronizationofthenetwork,itisnaturaltoassumethateachnodetriestosynchronizetoitsneighborsbyincreasingthecouplingstrengthfromthem.Wesupposethatthestrengthtoanodeifromallitskineighborsincreasesuniformlyamongthekiconnec-tions,inordertosuppressitsdifference ifromthemeanactivityofitsneighbors,namely,

_i i= 1 i ;V(2)Gij t AijVi t ;

P

where i jH xi ÿ 1=ki jAijH xj j,and >0isthe

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