Comparing Agent-Based and Differential Equation Mo

1HeterogeneityandNetworkStructureintheDynamicsofDiffusion:Compar[e-mail],FallsChurch,[e-mail]ement,CambridgeMA02142RevisionofAugust2007ForthcomingManagementScienceWethankRekaAlbert,JoshuaEpstein,RosannaGarcia,EdKaplan,DavidKrackhardt,MarcLipsitch,NelsonRepenning,PerwezShahabuddin,SteveStrogatz,DuncanWatts,LarryWein,membersoftheMIDASGroupand2006MIDASworkshop,theassociateeditorandreferees,andseminarparticipantsatMIT,the2004NAACSOSconferenceand2004InternationalSystemDynamicsConferenceforhelpfulcomments.VentanaSystemsandXJTechnologiesgenerouslyprovidedtheirsimulationsoftwareandtechnicalsupport.FinancialsupportprovidedbytheProjectonInnovationinMarketsandOrganizationsattheMITSloanSchool.2HeterogeneityandNetworkStructureintheDynamicsofDiffusion:ComparingAgent-BasedandDifferentialEquationModelsAbstractWhenisitbettertouseagent-based(AB)models,andwhenshoulddifferentialequation(DE)modelsbeused?WhereDEmodelsassumehomogeneityandperfectmixingwithincompartments,ABmodelscancaptureheterogeneityacrossindividualsandinthenetworkofinteractionsamongthem.ABmodelsrelaxaggregationassumptionsbutentailcomputationalandcognitivecoststhatmaylimitsensitivityanalysisandmodelscope.Becauseresourcesarelimited,thecostsandbenefitsofsuchdisaggregationshouldguidethechoiceofmodelsforpolicyanalysis.Usingcontagiousdiseaseasanexample,wecontrastthedynamicsofastochasticABmodelwiththoseoftheanalogousdeterministiccompartmentDEmodel.Weexaminetheimpactofindividualheterogeneityanddifferentnetworktopologies,includingfullyconnected,random,Watts-Strogatzsmallworld,scale-free,andlatticenetworks.Obviouslydeterministicmodelsyieldasingletrajectoryforeachparameterset,whilestochasticmodelsyieldadistributionofoutcomes.Moreinterestingly,theDEandmeanABdynamicsdifferforseveralmetricsrelevanttopublichealth,includingdiffusionspeed,peakloadonhealthservicesinfrastructureandtotaldiseaseburden.Theresponseofthemodelstopoliciescanalsodifferevenwhentheirbasecasebehaviorissimilar.Insomeconditions,however,thesedifferencesinmeansaresmallcomparedtovariabilitycausedbystochasticevents,parameteruncertaintyandmodelboundary.Wediscussimplicationsforthechoiceamongmodeltypes,focusingonpolicydesign.Theresultsapplybeyondepidemiology:frominnovationadoptiontofinancialpanics,manyimportantsocialphenomenainvolveanalogousprocessesofdiffusionandsocialcontagion.Keywords:AgentBasedModels,Networks,Scalefree,Smallworld,Heterogeneity,Epidemiology,Simulation,SystemDynamics,ComplexAdaptiveSystems,SEIRmodel3Spurredbygrowingcomputationalpower,agent-basedmodeling(AB)isincreasinglyappliedtophysical,biological,socialandeconomicproblemspreviouslymodeledwithnonlineardifferentialequations(DE).Bothapproacheshaveyieldedimportantinsights.Inthesocialsciences,agentmodelsexplorephenomenafromtheemergenceofsegregationtoorganizationalevolutiontomarketdynamics(Schelling1978;LevinthalandMarch1981;Carley1992;Axelrod1997;LomiandLarsen2001;Axtell,Axelrod,EpsteinandCohen2002;Epstein2006;Tesfatsion2002).Differentialanddifferenceequationmodels,alsoknownascompartmentalmodels,haveanevenlongerhistoryinsocialscience,includinginnovationdiffusion(Mahajan,MullerandWind2000)andepidemiology(AndersonandMay1991).WhenshouldABmodelsbeused,andwhenareDEmodelsappropriate?Eachmethodhasstrengthsandweaknesses.Theimportanceofeachdependsonthemodelpurpose.NonlinearDEmodelscaneasilyencompassawiderangeoffeedbackeffects,buttypicallyaggregateagentsintoarelativelysmallnumberofstates(compartments).Forexample,innovationdiffusionmodelsmayaggregatethepopulationintocategoriesincludingunaware,aware,inthemarket,adopters,andsoon(Urban,HauserandRoberts1990;Mahajanetal.2000).However,withineachcompartmentpeopleareassumedtobehomogeneousandwellmixed;thetransitionsamongstatesaremodeledastheirexpectedvalue,possiblyperturbedbyrandomevents.Incontrast,ABmodelscanreadilyincludeheterogeneityinindividualattributesandinthenetworkstructureoftheirinteractions;likeDEmodels,ABmodelscanbedeterministicorstochasticandcancapturefeedbackeffects.ThegranularityofABmodelscomesatsomecost.First,theextracomplexitysignificantlyincreasescomputationalrequirements,constrainingtheabilitytoconductsensitivityanalysis.Asecondcostofagent-leveldetailisthecognitiveburdenofunderstandingmodelbehavior.Linkingthebehaviorofamodeltoitsstructurebecomesmoredifficultasmodelcomplexitygrows.Finally,limitedtimeandresourcesforcemodelerstotradeoffdisaggregatedetailandthebreadthofthemodelboundary.Modelboundaryherestandsfortherichnessofthefeedbackstructurecapturedendogenouslyinthemodel(MeadowsandRobinson1985,Sterman2000).Forexample,anagent-baseddemographicmodelmayportrayeachindividualseparatelybutassumeexogenousfertilityandmortality;suchamodelhasanarrowboundary.Incontrast,anaggregatemodelmaylumptheentirepopulationintoasinglecompartment,butmodelfertilityandmortalityasfunctions4offoodpercapita,healthcare,pollution,normsforfamilysize,etc.,eachofwhich,inturn,aremodeledendogenously;suchamodelhasabroadbou