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Albert Scale-Free Networks

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Albert Scale-Free Networks

    Albert Scale-Free Networks Commun.Theor.Phys.(Bering,China)48(2006)PP.10111016

    @InternationalAcademicPublishersVo1.46,No.6,December15,2006 AModifiedEarthquakeModelBasedonGeneralizedBarabfisi-AlbertScale.Free Networks

    LINMin,1"WANGGang,

    2,3andCHENTian.Lun

    DepartmentofMathematics,OceanUniversityofChina,Qingdao266071,China InstituteofOceanology,theChineseAcademyofSciences,Qingdao266071,China GraduateSchool,theChineseAcademyofSciences,Beijing100049,China DepartmentofPhysics,NankaiUniversity,Tianjin300071,China (ReceivedJanuary17,2006)

    AbstractAmodifiedOlami-Feder-ChristensenmodelofselLorganizedcriticalityongeneralizedBarab,~si-Albert

    (GBA)scMe-fi-eenetworksjsinvestigated.Wefindthatourmodeldisplayspower-lawbehaviorandtheavalanche

    dynamicalbehaviorissensitivet0thetopologicalstructureofnetworks.Fhrthermore.theexponentofthemodel

    dependsonb,whichweightsthedistanceincomparisonwiththedegreeintheGBAnetworkevolution.

    PACSnumbers:05.65.+b,45.70.Ht

    Keywords:self-organizedcriticality,avalanche,GBAscale-freenetworks 1Introduction

    Manysocial,biological,andcommunicationsystems

    canbemodelledascomplexnetworks.Incomplexsys.

    tems,thenodesrepresentindividualsororganizations,

    andthelinksmodeltheinteractionamongthem.Re-

    cently,BarabiandAlberthavestudiedaclassoflarge growingnetworkswhosedegreedistributionfollowsa powerlawk.,JSincepower.1awisfreeofachar

    acteristicscale,thesenetworksarecalledscalefreenet. works.Scientistshavefoundthatmanyrea1.worldnet. workshavescalefreetopologicalstructuressuchasthe World-Wide-Web,socialacquaintancenetworks.biologi. calnetworks.

    AlthoughtheBarabsiAlbertfBA1scalefreenetwork

    isagoodrepresentationforahugevarietyofrealnetworks, itcannotreproducethecharacteristicsofthosesystems inwhichedgesarenotcostless(e.g.powergridsorneural networks).Toovercomethisproblem,Cosenzaeta1.in. troducedanewnetworkthatgeneralizestheBAscale.free network.Initialconditionandgrowingprocessofgener. alizedBarabksiAlbert(GBAforabbreviation1network arethesameasoftheBAnetwork.Theonlydifference isthatinthepreferentialattachmentGBAmbdeltakes int.

    oaccountthephysicaldistancebetweenaodes,which inmostrealcasesisanimportantparameterinthenet. workevolution.IF0rinstance.itismorelikelythata

    neuronconnectstonearbyneuronsinaneuralnetwork. GBAscale-freenetworkhasaprecisespatialarrangement. Suchanetwork.becauseofitsplausibilitybothinstatic characteristicsandinthedynamicalevolution.isagood representationforthosereanetworkswhoseedgesarenot

    costless.Is]

    Anotherareaofactiveresearchisrelatedtosystems thatpresentself-organizedcriticality(soc).Theconcept

ofself-organizedcriticalityreferstocertaindissipativesys

    tems,withmanydegreesoffreedom,naturallyevolveto acriticalstatecharacterizedbypower.1awdistributionin spaceandtime_I4JOneofthesystemshavingbeenstud. iedinconnectionwithSOCisOlamiFederChristensen

    (OFC)modelforearthquakes.Recently,someworkshave beendonetoinvestigatehowthetopologicalstructureof networkaffectsS0CbasedonOFCmode1.Asiswell known,OFCmodelsofSOConaregularlatticeandon arandomgraphhavebeeninvestigated.,.Amodified

    earthquakemodelofSOCbasedonsmallworldnetworks hasbeendiscussedinourpreviousworks.[7I8Zhouet

    introducedanalternativemodeltomimicthecatastro- phesinthescale-freenetwork.【引Naturally,peoplewillask.

    whetherSOCbehaviorscanbedisplayedinOFCmodel whosenetworktopologicalarchitectureisGBAscale.free network.So,basedonOFCearthquakemodel,whichisa typicalSOCmode1.weintroduceamodifiedearthquake modelinGBAscale-freenetwork.Inourmode1.thedis- tributionofavalanchesizeshowspowerlawbehaviorand theexponentrdependsonb,whichweightsthedistance incomparisonwiththedegreeintheGBAnetworkevo- lution.

    2TheModel

    2.1StaticPropertiesoftheGraph

    AgenericnetworkisagraphG,inwhichtheNnodes 'TheprojectsupportedbyNationalNaturalScienceFoundationofChinaunderGrantNo.902

    03008andtheDoctoralFoundationof

    theMinistryofEducationofChina

    tE.mail:linminmin~eyou.com

    1012LINMin,WANGGang,andCHENTian?LunVlnl_46 representthebasiccomponentofthenetworkandtheK edgesrepresentaninteractionbetweenthem.?,erep,

    resentanetworkthroughitsadjacencymatrixA=(aij), whoseelementaijis1ifthereisanedgeconnectingnodes iand,and0otherwise.Thenetworkisundirectedand hasanaveragedegreeequalto(k)2KIN.

    Tocharacterizethestructuralpropertiesofagraph. weusetwoparametersasdefinedinRef.f101.Thefirstis thecharacteristicpathlength:

    ,

    ?{?j?Gdij

    山两=,

    wheredisthelengthoftheshortest

    nodesiandjf.Thesecondparameter

    coecientC:

    c=N

    with

    Ci=numberofedgesin

    ki(ki——i)/2

    (1)

    pathconnecting

    istheclustering

    (2)

    (3)

    wherek~(ki——I)/2isthetotalnumberofpossibleedgesin Gi,whichissubgraphofthefirstneighborsofanodei. Thestructuralpropertiesdonottakeintoaccountthe rolethatphysicaldistanceplaysintheformationofnew

edgesbetweennodes.Toevaluatehowmuchthebuild

    ingcostofanetworkis.weuseacostparameter.3JThe

    structuralcostofthenetworkdefinedas

    Cost;?%?2(4)

    .

    ij6G

    whereaqistheelementoftheadjacencymatrixand

    loistheEuclideandistancebetweennodesiandJ. 2.2GeneralizedBarabdsi-Albert(GBA)Scale- e?etworks

    Wlegeneratethenetworkfollowingtheprescriptionof Cosenzaeta1.inRf31.Weconsidernodesplacedona two-dimensionalregularlatticewithfreeboundarycondi. tions.Butforfurtherdiscussion,wealsouseopenbound. aryconditionsinsubsection3.3.Thealgorithmbehind theGBAmodelisthefollowing.

    (i)Startwithasmallnumber(m0)ofnodes

    (ii)Thenweaddanewnodewithm(m0)edges

    (thatwillbeconnectedtothenodesalreadypresentinthe system).Theprobabilityforanewnodeitobeconnected withanalreadypresentnodeJis

    ?():鲁?^?Gkh/Z,~h'(5)

    wherekhisthedegreeofnodeh,lihistheEuclideandis- tancebetweennodesiandh,andbisanexponentthat weightsthedistanceincomparisonwiththedegree. (iii)Repeatstep(ii)untilthenumberofnodesisN, i.e.LxL.

    Inthisway,theconstructionofGBAmodelisfin

    ished.Thesituationb0correspondstotheoriginalBA

    scalefreenetwork.AnexampleofaGBAnetworkwith N=100nodesandm0=m=2underfreeboundary

    conditionsisshowninFig,1.

    Fig.1GeneralizedBarab~siAlbert(GBA)scalefree

    networkforN=100nodes.mo=m=2andtheexpo-

    nentb=1withfreeboundaryconditions.

    ?

    .

    .

    (c)-

    ????

    02468

    b

    Fig.2GeneralizedBarab~si-Albertscale-freenetwork forN=100nodes,mo=m=2.(a)Characteristicpath lengthL;(b)ClusteringcoefficientG;(c)Structuralcost, versustheexponentb.

    TocharacterizeGBAmodel.wehavecalculatedthe characteristicpathlengthL,theclusteringcoefficient, andthestructuralcost,asfunctionsofbfornetworkswith N=100andm0:m=2.InFig.2.characteristicpath lengthLincreaseswiththeincrementoftheexponentb. ClusteringcoemcientCcauseasignificantincreasefrom b=0tob=4.Moreover.whenweincreasetheexponent o0

No.6AModifiedEarthquakeModelBasedonGeneralizedBarab~si-AlbertScale-FreeNet

    works1O13

    b,long-rangeconnectionsarehamperedand,consequently, thecostdecreasesfromb0tob=4.

2,3DynamicsintheGjE}ANetwork

    Eachsiteofthelatticeisassociatedwitharealvari. able,whichisinitializedwitharandomvaluebetween 0andathresholdvalueFth=1.ThenwecaD.describe thedynamicalprocessofourmodelasfollows. (i)Findoutthemaximalvalueofall,Fmax,and addFthFmaxtoallsites.Whenoneofthemreachesthe thresholdFth=1,itbecomesunstable.Atthispoint,an avalanche(earthquake)starts.

    (ii)IfanyFththenredistributetheenergy

    onthei-thsitetoitsneighbors:

    _?乃+/,_?0,(6)

    forallnodesJadjacenttoi,whereisthedegreeofnode

    i.

    (iii)Repeatstep(ii)untilallsitesarestable.Define thisprocessasoneavalanche.

    (iv)Beginstep(i)againandanothernewavalanche begins.

    TheparameterQcontrolsthelevelofconversationof thedynamics.WhenQ=1.themodelisconservative. Otherwise,themodelisnonconservativeforQ<1.Let usfinallydiscusstheboundaryconditions.Theboundary beingfreemeansthattheblocksintheboundarylayerare connectedonlytoblockswithinthefault.5]Inourmode1.

    weuseQbc=Q/(+1Q),exceptatcornersiteswhere

    Qbc.c=a/(ki+22a).Theboundarybeingopenmeans

    thattheblocksintheboundarylayerarecoupledtoan imaginaryboundaryblockbysprings.5]Inourmode1.we

    useQ6c:Q'/kt.

    3SimulationResults

3.1Power-LawBehavlorandInfluenceD,the

    E~rponentb

    HereweuseaGBAnetworkofthesize20x20,where Q=0.98,mom=2arefixed.Thenwechangeb,our

    aimistoinvestigatethedistributionoftheavalanchesize fordifferentb.

    ToprovetheSOCofoursystem,wemeasuretheprob- abilitydistributionoftheavalanchesizes.TheEuclidean 1atticementionedinFig.3isatwo-dimensionalsquare latticeandBAnetworkisGBAnetworkwithb:0.

    mom=2underfreeboundaryconditions.Thusthe averagedegreeofthenetworksis(k)4.Asshownin Fig.3,thedistributionofavalanchesizeshaspower-law behaviors,P(S)S_.,whichisregardedas"fingerprint" forSOC.11JSimilartotheresultinRef.f91,thedistribu- tionofavalanchesizeinBAnetworkfollowsastraight lineformorethanthreedecades.Withtheincrementof b,theprobabilityoflargesizeavalanchesoccurringde- creasesandthecutoffintheavalanchesizedistribution decreases.?thinkthatthebehaviorsarecausedbythe exponentbintheGBAnetwork.Thegreatertheexpo- nentbisgreatertheimportanceofdistance(incompari- sonwiththedegree)intheGBAnetworkevolution.For b=0,therearemanylong-rangeconnectionsinthenet- work.AsshowninFig.2.forincreasingvaluesoftheex- ponentb,thecharacteristicpathlengthandtheclustering coe~cientincreases.andthecostdecreases.Thenetwork; tendstowardshomogeneityandthenumberoflong-range connectionsdecreases.Forlatticecase.thereisnopres. enceoflong.rangeconnections.Withtheincrementof

    b.thenumberoflong-rangeconnectionsinthenetwork decreases.Sotherangesoftheparticularsites'localin. teractionarereduced.Thelargesizeavalancheshaveless probabilitytoOccurandthecutoffintheavalanchedis.. tributiondecreases.ThemaximalavalanchesizeinGBA networkisgreaterthanthatinEuclideanlattice.

    S

    Fig.3TheprobabilityoftheavalanchesizeP(S1as afunctionofSwithsystemsizeL=20.d=0.98for GBAnetworkwithb=0,4,andEuclideanlattice,re- spectively.

    Atthesainetime,wepresentthedependenceofthe exponentTontheexponentb.InFig.4(a),ascanbe seen.thevalueof7-decreaseswiththeincrementofb.W alsoinvestigatetherelationbetweentheaverageavalanche size(S)andtheexponentbinFig.4(b).Theaverage avalanchesize(S)alsodecreasesasbincreases.There- sultisconsistentwiththeconclusionmentionedabove. InFig.5,wedrawthemeansize(S)ofavalanches originatedfromnodesoftheGBAnetworkwiththesame degreeforsystemsizeL=20,Q=0.98withb=3and8, respectively.Overall,(S>b:3>(S>b:8forthesamedegree

    l0l4LINMin,WANGGang,andCHENTian-LunV_0i.46 ofnodes.Itaccordswiththeresultthat(S)forb:3iS largerthan(S)forb=8inFig.4(b).

    InFig.2,whenbisequalto3,theGBAnetworkhas lowcharacteristicpathlengthandhighclustering.The power-lawdegreedistributionofGBAnetworkwithb=3

isbetterpreserved.3lMoreover,forb=3,theGBAnet.

    workalsomeetstherequirementsoflowcost,whichisfun

    damentalforreal-worldnetworks.FromFig.4(a),wecan seethedistributionofavalanchesizeinb=3GBAnet. workfollowspowerlawbehavior.Inthefollowing.b=3 willbeused.

    

    Fig.4a)Thepower-lawexponentroftheavalanche sizedistribution;(b)Theavalancheaveragesize(S)aSa functionof6forL=2O.=0.98.

    (s)

    nodedegree

    Fig.5Meansize(S)ofavalanchesoriginatedfrom nodeofthenetworkwiththesamedegreeforL=2O, =0.98withb=3andb=8.respectively.

    3.2Influenceo|the~attieeSize

    Intheor.

    iginalOFCmodel,thelatticesizeisimpor

    tant.Wealsoinvestigatetheeffectofthelatticesizeon SOCbehaviorinthemode1.Ifthelargesizecutoffof theavalanchedistributiondoesnotscalewiththesystem size.itisalocalizedbehavior.otherwiseitmeansSOC.tsJ InFig.6,weshowtheresultsofsimulationswithb=3, =0.98forL=20,30and50,respectively.Wecallsee thatthelargesizecutoffintheavalanchesizedistribution scalewiththesystemsize.whichisindicativeofacritical state.5]

    S

    Fig.6TheprobabilityoftheavalanchesizeP(S)a8a functionofsizeSfor=0.98.b=3withdifferentsize

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