磁性流體中的一些問(wèn)題課件

Disconnected-connectednetworktransitionsandphaseseparationdrivenbycoevolvingdynamics由耦合演化驅(qū)動(dòng)的網(wǎng)絡(luò)結(jié)構(gòu)與相分離行為

PakMingHui許伯銘

DepartmentofPhysicsTheChineseUniversityofHongKong香港中文大學(xué)物理系Incollaborationswith:OliverGr?ser顧皓森

(CUHK)ChenXU許晨

(SoochowUniversity)CCCN2010(15-17October2010,Suzhou)Disconnected-connectednetworkDynamicmodels(SIS,SIR,opinionformation),orgames(PD,SG,…)NETWORKS(groupdynamics)NEWFEATURES?COMPUTERSIMULATIONSTHEORIESREALSYSTEMSCOEVOLVINGSYSTEMTwodynamicsinfluencingoneanotherToreadmoreonthetopicingeneral:PercandSzolnoki,Biosystems99,109(2009)SzaboandFath,PhysicsReports446,97(2007)GrossandBlasius,J.R.Soc.Interface5,259(2008)Dynamicmodels(SIS,SIR,opiniDynamicmodels(SIS,SIR,opinionformation),orgames(PD,SG,…)NETWORKS(groupdynamics)NEWFEATURES?COMPUTERSIMULATIONSTHEORIESREALSYSTEMSCOEVOLVINGSYSTEMTwodynamicsinfluencingoneanotherThegeneralideashavebeenappliedto:Adaptiveepidemicmodels:e.g.,Grossetal.,PRL96,208701(2006);ShawandSchwartz,PRE71,066101(2008)Opinionformationmodels:e.g.,Vazquezetal.,PRL100,108702(2008);Nardinietal.,PRL100,158701(2008)Warsandhumanconflicts:e.g.Bohorquezetal.,Nature462,911(2009);Zhaoetal.,PRL103,148701(2009)Dynamicmodels(SIS,SIR,opiniDynamicmodels(SIS,SIR,opinionformation),orgames(PD,SG,…)NETWORKS(groupdynamics)NEWFEATURES?COMPUTERSIMULATIONSTHEORIESREALSYSTEMSCOEVOLVINGSYSTEMTwodynamicsinfluencingoneanotherAndmore…(fromPMHui’sgroup):Modelingofguildsinonlinegames(WorldofWarcraft)andLAstreetgangs–Zhaoetal.,PRE79,066117(2009)Effectsofsocialgroupdynamicsoncontagion(YouTubedownloads,foreignexchangerates,flu)–Zhaoetal.,PRE81,056107(2010)Dynamicmodels(SIS,SIR,opiniCo-evolvingModeling–“JobHuntingModel”AnagentlooksforagroupthathethinkshecouldcontributeAgroupassesstheagenttoseeifhecancontributetothegroupAfterjoininggroup,agenthasabetterunderstandingofthegroupandassessthegroup(CanIreallycontribute?)Ifagentisunhappywiththegroup,agentwillquit!IfagentfeelsOKwiththegroup,hestillwantstofindabettergroupIfhefindsabettergroup,hewillswitchgroup;ifnot,hestaysTeamformationmodel(agentswithskillsthatcomplementeachother)againstkinship(buddy-buddy)modelCo-evolvingModeling–“JobHuMainEmpiricalResultsfromDataSets:OnlineguildsandOfflinestreetgangsWowGuildsizedistributionN(s)forallguildsin3serversS1,S2,S3(puttogether)inOct2005Totalplayers:76686CumulativesizedistributionInset:ChurnvsguildsizeCumulativegangsizedistributionofLAStreetgangswithallethnicityputtogetherTotalmembers:5214SmalldatasetsStepseveninN(s’>s)Datafrom:DucheneautandYee(PaloAltoResearchCenter)MainEmpiricalResultsfromDaWoWEmpiricaldata(blue)&Team-formationModelingResults(red)CumulativeguildsizedistributionandChurnvsguildsizeNfromdataistakenasinput(datainOct2005)N=76686N=24033N=24477N=28176WoWEmpiricaldata(blue)&TeCumulativegangsizedistributionData(blue)andteam-formationmodelingresults(red)Dashedline(kinship/”buddy-buddy”model)N=5214SeeAPSNewsitem(June2009)/synopsis-for/10.1103/PhysRevE.79.066117forannewsitemreportingourworkCumulativegangsizedistributHere,weuseanadaptivesnowdriftgameasanexampletoillustrate…--howcoupleddynamicsinfluenceeachotherandexplicitcoupledtransitionsintheformofdisconnectedtoconnectednetworktransition(structural)highlycooperativetolowercooperativepopulation(functional)segregatedphasetomixed-characterphase(populationcharacteristics)frozentocontinouslyevolving(dynamical)--howonecouldapproachsuchproblemsanalytically--whattolookatinformulatingatheoryanditsvalidity--whatapropertheorycaninformusaboutthepropertiesofthesystemHere,we

SnowdriftGame(SDG)[1]TwodriversheadinghomeinoppositedirectionsBlockedbyasnowdriftEachdriver:2actions/characters

C(“cooperate”)=toshovelthesnowdriftD(“not-to-operate”)OR“defect”(inprisoner’sdilemmalanguage)=nottoshovelScenario:[1]J.M.Smith,EvolutionandtheTheoryofGames(CambridgeUniv.Press1982).Inothercontexts,the“gameofchicken”.SnowdriftGame(SDG)[1]Twodb=rewardofgettinghomec=cost(doingthelaboriousjobofshoveling)b>c>0Player1Player2CCDDSuckerpayoffb=rewardofgettinghomeb>c>0Player2CCDDb>c>0definesthesnowdriftgameItfollowsthatT>R>S>P(definesSDG)Player1Player2CCDDb>c>0defineShowingonlythepayoffsofplayer1:CCDDRTSPSnowdriftGame:T>R>S>PPrisoner’sDilemma:T>R>P>SDifficulttomeasurepayoffsaccuratelySDGisanalternativetoPDinstudyingcooperationincompetingpopulationsShowingonlythepayoffsofOften,useoneparameterrtorepresentthepayoffs:T>R>S>P(=0)1+r>1>1-r>0(0<r<1)CCDD11+r1-r0SDG[largerr(temptation)tendstopromoteD-character]Often,useoneparameterrtPC,switchNothinghappensPD,switchPC,rewirePD,rewirePC,switch=r/2PD,switch=(1-r)/2PC,rewire=1-r/2PD,rewire=(1+r)/2Howtoassignswitchingprobabilities(CSandDSevents)?

Dissatisfaction!Doesn’tmeetexpectation,thusrational!CDPC,switchNothinghappensPD,swiExpectation: WhenIplayC,Iexpecttoget1(opponentisC) WhenIplayD,Iexpecttoget1+r(opponentisC)Thus,whenopponentisC,received=expected =>noincentivetomakeanychangesThus,dissatisfactioncomesinonlywhenopponentplaysD =>switchcharacterorrewiringS=expectedpayoff–receivedpayoff=P(α,C)-P(α,D)Wedefineaparameter,calledthedisappointmentS,whenopponentplaysDasSwitchingProbabilityP

S→P=β

SIfnotswitched,cutlinkandrewiretosomeoneelse.(Here,wetakeβ=1/2)Node-drivendynamicsCD-linksANDDD-linksaretheactivelinks(possiblesystemevolution)Expectation: WhenIplayC,PC,switch=r/2PD,switch=(1-r)/2PC,rewire=1-r/2PD,rewire=(1+r)/2Probabilitiesforthe4adaptiveeventsthatleadtosystemevolution“DissatisfiedAdaptiveSnowdriftGame”(DASG)PC,switch=r/2PD,switch=(1-r)Howdoesthelevelofcooperation(longtimebehavior)varywithr?Howdoesdissatisfactionbehavioralternetworkstructure?Timeevolution?Constructinganalyticapproaches?Newfeatureshintedatbytheory?HowdoesthelevelofcooperatInitially,wehave50%cooperatorsrandomlydistributedinthelattice.Theresultsindicatetworegimeswithdifferentfeatures.Whatiftheinitialfrequencyofcooperatorsisvaried?Thisfiguregivesusamessageoftheextentofcooperation.

Buthowthedifferentcharactersconnected?

Definition:fC=numberofC-nodes/numberoftotalnodesInitially,wehave50%cooperlCClCDlDDInitially,wehave10%cooperatorsrandomlydistributedinthelattice.Thesymbolsshowatransitionbehavioratsomevalueofr.Canweobtainthefeaturesofthepreviousfigurebasedonthelinkdensities?Thisfiguregivesusamessageoflinkdistributionsonnetwork.CDCDDefinition:lxy=numberofXY-links/numberoftotallinkslCClCDlDDInitially,wehave1DisconnectedConnectedDisconnected-connectednetworktransitionaccompanyingaC/Dphase-separatedandmixedphasetransitionRef:Graser,Xu,Hui,EPL87,38003(2009)DisconnectedConnectedDisconnecLow-rPhaseHigh-rPhaseLevelofcooperationNetworkStructureDynamicsPopulationHighDisconnectedFrozenSegregatedLowConnectedEvolvingMixed-characterLow-rPhaseHigh-rPhaseLevelor=0.1r=0.9fci=0.1fci=0.9(a)&(b):Initialfrequencyofcooperationfci=0.(c)&(d):Forcost-to-benefitratior=0.3.Moresimulationresults—trajectoriesshowingtimeevolutionDefinition(x-axis):NC-NDm=

NTrajectoryofSystems(timeevolution)r=0.1r=0.9fci=0.1fci=0.9(a)&Constructingatheory…Recall:ThenumberofCooperators(i.e.,NC

orfC

orm)determinesthefractionofcharactersonnodes.Thenumberoflinks(i.e.,LCC…orlCC…orml)indicates,onaverage,howlinksaredistributedbetweennodes.lCClCDlDClDDCDWhenanadaptiveeventoccurs,localenvironmentofupdatingnodechanges,leadingtocorrespondingchangesinvariablessuchasnode/linknumbers.Constructingatheory…Recall:Writedownthechangeofthevariablesingeneral(recall:CDandDDareactivelinks):Probabilitythatthenodeinactiontakesoncharactern=CorDConditionalprobabilityofhavingκlinksaroundanodeofcharacternProbabilitythatamongκlinksthereareλndlinksFractionofnd-linksamongκ

links(prob.ofpickinganactivelink)ProbabilitythataneventEoccurs(switch/rewire)undertheconditionthatthenodeisofcharactern？(＊)Remark:Couldstartfromnode-levelequationsandconstructequationsforglobalquantitiesWritedownthechangeofthevDefinitions:C→DM=NC-ND,Ml=LCC-LDDD→CCD-cut&rewireDD-cut&rewire-2200λCC-λCDλDD-λCD-1+ND/NNC/N-λCC-λCDλDD+λCDNC/NNC/N?M?LCD?MlCDND/NNC/NDefinitions:C→DM=NC-ND,Ml=LCToillustratethedifferentwaysofmean-fieldtreatment,wewritedownoneoftheequationsbasedon(＊):Here,wechoosefC,lCD,lCCasindependentvariables.<…>denotesanaverageoveratypeofnodes(subscript).Note:Theseaveragesdistinguishdifferenttypesofnodes. NeedtotreatCandDnodesseparately.ToillustratethedifferentwaToproceed,wefirstdecouplethequantityoftheformWetreatthefirstmomentsusingglobalmeanvalues,i.e.,Toproceed,wefirstdecoupleAim:ToclosethesetofequationsThereareseveralwaystotreatthesecondmoments.(1)SimpleSquaredClosure(SSC)[simplestapproximation]Secondmomentsassumedtobeequaltothefirstmomentssquared.PhysicalPicture:EveryCnodehasidenticalneighborhood,everyDnodehasidenticalneighborhood;butCandDcouldhavedifferentneighborhoods(thusignoredfluctuations).Aim:ToclosethesetofequatAclosedsetof3equations–doesitwork?Lines:FromclosedsetofequationsusingSSCanditeratethemintimeCaptureallkeyfeaturesqualitatively,includingnon-monotonicbehavior!Ref(DASG):Graser,Xu,Hui,EPL87,38003(2009)Aclosedsetof3equations–Theorycapturesthedisconnected-connectednetworktransitionandphaseseparationfci=0.1(squares),0.7(circles),0.9(triangles)TheorycapturesthedisconnectHowaboutthetrajectories?Lineshowsthelocationsofendpointsascalculatedbyiteratingclosedsetofequationstolongtime.Withinitialfrequencyofcooperationfc=0.Howaboutthetrajectories?LinTimeevolutionofdegreesfci=0.8andr=0.8fci=0.1SteadystatemeandegreesLinesareresultsofequations,symbolsaresimulationresultsNotethenecessityintreatingCandDnodesseparatelyTimeevolutionofdegreesfci=0Whilesimpletheorycapturesallthemainfeatures,therearediscrepanciesbetweensimpletheoryandnumericalresults!Canweimprovethetheory?Bettermomentclosureschemes?WhilesimpletheorycapturesaAlternativewaystotreatthesecondmomentsandtoclosetheequations(2)Binomialdistributiontreatment(BINO)Picture:AssumethatC-node(orD-node)ofdegreeκ

hasabinomialdistributionoflinksλCD.ComparingwithSSCtreatment,wehaveextratermsinBINOtreatment!Alternativewaystotreatthe(3)Aside:Keeling-Eamestreatment(KE)Fluctuationsareincludedinawaythatassumesthevarianceequalstothemean.ComparingwithSSCtreatment,thereisanextraterminKEtreatmentAmomentclosuremethodusedbyGrossetal.inPRL96,208701(2006)foranadaptiveSISmodel,basedonKeelingandEamesPNAS99,13330(2002)andJ.R.Soc.Interface2,295(2005)forepidemicmodelsButKEapproximationturnsouttobeabadapproximationforthepresentmodel.(3)Aside:Keeling-EamestreatSSCtreatment(dashedlines)BINOtreatment(solidlines)–WorkedmuchbetterinthedisconnectedstateComparingsimpleandmodifiedtheorieswithnumericalresultsRef:OliverGraser,PhDDissertation(CUHK2010)SSCtreatment(dashedlines)BIFixedPointAnalysisFixedPoints:?Nc=?LCC=?Lcd=0Obviouscandidate:LCC=κ/2,Lcd=0(nodissatisfiedlinks–disconnectedstate)Occursforsmallr,largefciFixed-pointanalysisiscarriedoutbasedonMFequationsusingabinomialapproximationFixedpointvsiterationofequationsstartingfromsomeinitialcondiitionsTakingadvantageofhavingasetofmean-fieldequationsFixedPointAnalysisFixedPoinFixedPointAnalysisFixedPoint:?Nc=?LCC=?Lcd=0Obviouscandidate:LCC=κ/2,Lcd=0(nodissatisfiedlinks)Stickyfixedpoint(disconnectedstate)!Occursforsmallr,largefciConfirmedinsimulations

OthertypeoffixedpointswithLcd>0(connectedandmixedphasestate)existsforlargerr.Whataboutsmallr?FixedPointAnalysisFixedPoinFixedPointAnalysisFixedPoint:?Nc=?LCC=?Lcd=0Obviouscandidate:LCC=κ/2,Lcd=0(nodissatisfiedlinks)Stickyfixedpoint!Occursforsmallr,largefciCanbeconfirmedinsimulations.OthertypeoffixedpointwithLcd>0(connectedmixed-phasestate)existsforlargerr.Whataboutsmallr?Rootsofmean-fieldequationsindicateafixedpointofconnectedmixed-phasestatealsoexistsforsmallr!Ifexists,thisisanUNUSUALstate,ascooperationDROPSasTEMPTATIONtobeuncooperativeDROPS!Whynotfoundinsimulations?Unstableorstablefixedpoint?FixedPointAnalysisFixedPoinStabilityoffixedpointAfixedpointisattractiveiftrajectoriesstartinginasmallenvironmentaredrawntowardsit.LinearizationofthesystemandlookingateigenvaluesofJacobianMatrix.IftheJacobi-Matrixhasonly(real)negativeeigenvalues,thenthefixedpointsareattractive.Thefixedpointhasnegativeeigenvalues(twoareidentical,approachingzeroasrdrops,andonebecomesincreasinglynegative),thusitisstable!Howcanwerealizetheconnectedmixed-phaseinsmallr?StabilityoffixedpointAfixeRealizingtheunusualstateWhynotfoundinsimulations?Relatedtochoiceofinitialcondition!Tracesystemfromr=1.EquilibrateRecordconfiguration(useitasnextinitialcondition)ReducerObservedconnectedmixed-phasestateforlowervaluesofr(butbasinshrinksasrdecreases)Systemfreezesatr=0.05

Unusualconnected