Markov Chain Aggregation for Agent-Based Models PDF
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Understanding Complex Systems Sven Banisch Markov Chain Aggregation for Agent-Based Models Springer Complexity Springer Complexity is an interdisciplinary program publishing the best research and academic-level teaching on both fundamental and applied aspects of complex systems – cutting across all traditional disciplines of the natural and life sciences, engineering, economics,medicine,neuroscience,socialandcomputerscience. Complex Systems are systems that comprise many interacting parts with the ability to generate anew qualityof macroscopic collectivebehavior themanifestations of whichare the spontaneous formation of distinctive temporal, spatial or functional structures. Models of such systems can be successfully mapped onto quite diverse “real-life” situations like theclimate,thecoherentemissionoflightfromlasers,chemicalreaction-diffusionsystems, biologicalcellularnetworks, thedynamicsofstockmarketsandoftheinternet,earthquake statistics and prediction, freeway traffic, the human brain, or the formation of opinions in socialsystems,tonamejustsomeofthepopularapplications. Although their scope and methodologies overlap somewhat, one can distinguish the following main concepts and tools: self-organization, nonlinear dynamics, synergetics, turbulence,dynamicalsystems,catastrophes,instabilities,stochasticprocesses,chaos,graphs and networks, cellular automata, adaptive systems, genetic algorithms and computational intelligence. ThethreemajorbookpublicationplatformsoftheSpringerComplexityprogramarethe monograph series“Understanding ComplexSystems”focusing on thevariousapplications of complexity, the “Springer Series in Synergetics”, which is devoted to the quantitative theoreticalandmethodological foundations,andthe“SpringerBriefsinComplexity”which are concise and topical working reports, case-studies, surveys, essays and lecture notes of relevance to the field. In addition to the books in these two core series, the program also incorporatesindividualtitlesrangingfromtextbookstomajorreferenceworks. EditorialandProgrammeAdvisoryBoard HenryAbarbanel,InstituteforNonlinearScience,UniversityofCalifornia,SanDiego,USA DanBraha,NewEnglandComplexSystemsInstituteandUniversityofMassachusettsDartmouth,USA Péter Érdi, Center for Complex Systems Studies, Kalamazoo College, USA and Hungarian Academy ofSciences,Budapest,Hungary KarlFriston,InstituteofCognitiveNeuroscience,UniversityCollegeLondon,London,UK HermannHaken,CenterofSynergetics,UniversityofStuttgart,Stuttgart,Germany Viktor Jirsa, Centre National de la Recherche Scientifique (CNRS), Université de la Méditerranée, Marseille,France JanuszKacprzyk,SystemResearch,PolishAcademyofSciences,Warsaw,Poland KunihikoKaneko,ResearchCenterforComplexSystemsBiology,TheUniversityofTokyo,Tokyo,Japan ScottKelso,CenterforComplexSystemsandBrainSciences,FloridaAtlanticUniversity,BocaRaton,USA Markus Kirkilionis, Mathematics Institute and Centre for Complex Systems, University of Warwick, Coventry,UK JürgenKurths,NonlinearDynamicsGroup,UniversityofPotsdam,Potsdam,Germany AndrzejNowak,DepartmentofPsychology,WarsawUniversity,Poland HassanQudrat-Ullah,SchoolofAdministrativeStudies,YorkUniversity,Canada LindaReichl,CenterforComplexQuantumSystems,UniversityofTexas,Austin,USA PeterSchuster,TheoreticalChemistryandStructuralBiology,UniversityofVienna,Vienna,Austria FrankSchweitzer,SystemDesign,ETHZurich,Zurich,Switzerland DidierSornette,EntrepreneurialRisk,ETHZurich,Zurich,Switzerland StefanThurner,SectionforScienceofComplexSystems,MedicalUniversityofVienna,Vienna,Austria Understanding Complex Systems FoundingEditor:S.Kelso Future scientific and technological developments in many fields will necessarily dependuponcomingtogripswithcomplexsystems.Suchsystemsarecomplexin boththeircomposition–typicallymanydifferentkindsofcomponentsinteracting simultaneouslyandnonlinearlywitheachotherandtheirenvironmentsonmultiple levels–andintherichdiversityofbehaviorofwhichtheyarecapable. TheSpringerSeriesinUnderstandingComplexSystemsseries(UCS)promotes new strategies and paradigms for understanding and realizing applications of complex systems research in a wide variety of fields and endeavors. UCS is explicitlytransdisciplinary.Ithasthreemaingoals:First,toelaboratetheconcepts, methodsandtoolsofcomplexsystemsatalllevelsofdescriptionandinallscientific fields,especiallynewlyemergingareaswithinthelife,social,behavioral,economic, neuro-andcognitivesciences(andderivativesthereof);second,toencouragenovel applicationsoftheseideasinvariousfieldsofengineeringandcomputationsuchas robotics,nano-technologyandinformatics;third,toprovidea singleforumwithin whichcommonalitiesanddifferencesintheworkingsofcomplexsystemsmaybe discerned,henceleadingtodeeperinsightandunderstanding. UCS will publish monographs, lecture notes and selected edited contributions aimedatcommunicatingnewfindingstoalargemultidisciplinaryaudience. Moreinformationaboutthisseriesathttp://www.springer.com/series/5394 Sven Banisch Markov Chain Aggregation for Agent-Based Models 123 SvenBanisch MaxPlanckInstituteforMathematics intheSciences Leipzig Germany ISSN1860-0832 ISSN1860-0840 (electronic) UnderstandingComplexSystems ISBN978-3-319-24875-2 ISBN978-3-319-24877-6 (eBook) DOI10.1007/978-3-319-24877-6 LibraryofCongressControlNumber:2015956382 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com) DieMathematikisteine KarikaturderWirklichkeit (PhilippeBlanchard) Preface Agent-based modeling is an interesting tool. It provides model developers with a greatdegreeoffreedomforthedesignofsystemsin whichheterogeneousentities interact with each other and the environment. Agent-based models (ABMs) are thereforea great tool to explorehow differentassumptionsabouthow individuals behaveand interactaffectthe evolutionof social, economicor ecologicalsystems asawhole. Themathematicalformalizationofthese models,however,isstillinits infancy partly due to the fact micro-level heterogeneity or complex interaction structures oftenleadtoeffectsinthesystemdynamicswhicharenoteasilyaccountedforby macroscopicformulationsoftheproblem.Toaddressthisissueanunderstandingof the transition from the most informative “atomic” level to the levels at which the system behavior is typically observed is important, because it can help to derive and evaluate models on specific levels, on the one hand, and to understand the temporaland spatial patterns that may emerge in that transition on the other. The bookathanddevelopsaMarkovchainapproachthatallowsarigorousanalysisofa classofmicroscopicmodelswhichspecifythedynamicsofacomplexsystematthe individuallevel.Itprovidesageneralframeworkofaggregationinagent-basedand relatedcomputationalmodelsbymakinguseoflumpabilityandinformationtheory inordertolinkbetweenthemicroandmacrolevelsofobservation. The starting pointis a microscopicMarkovchain descriptionof the dynamical process in complete correspondence with the dynamical behavior of the ABM, which is obtained by considering the set of all possible agent configurations as the state space ofa hugeMarkovchain.Thisis referredto asmicrochain,andan explicitformalrepresentationincludingmicroscopictransitionratescanbederived for a class of models by using the random mapping representation of a Markov process. The explicit micro formulation enables the application of the theory of Markovchainaggregation—namely,lumpability—inordertoreducethestatespace ofthemicrochainandrelatemicroscopicdescriptionstoamacroscopicformulation ofinterest.Well-knownconditionsforlumpabilitymakeitpossibletoestablishthe caseswherethemacromodelisstillMarkov,andinthiscaseweobtainacomplete vii viii Preface pictureofthedynamicsincludingthetransientstage,themostinterestingphasein applications. Forsuchapurposeacrucialroleisplayedbythetypeofprobabilitydistribution usedtoimplementthestochasticpartofthemodelwhichdefinestheupdatingrule and governs the dynamics. Namely, if we decide to remain at a Markovian level, thenthepartition,orequivalently,thecollectivevariables,usedtobuildthemacro model must be compatible with the symmetries of the probability distribution !. Microscopicheterogeneityandconstraintstranslateintodynamicalirregularitiesin themicrochainandrequirearefinementoftheaggregationandthecorresponding levelofobservation.Thisunderlinesthetheoreticalimportanceofhomogeneousor completemixingintheanalysisof“voter-like”modelsatuseinpopulationgenetics, evolutionarygametheoryandsocialdynamics. TheproblemofaggregationinABMsandthelumpabilityconditionsinparticular canbe embeddedintoa moregeneralframeworkwhichmakesuseof information theoryinordertoidentifydifferentlevelsandrelevantscalesincomplexdynamical systems. Lumpability and, respectively, the existence of a higher-levelMarkovian descriptionisoneoutofseveralmutuallyrelatedcriteriawhichaclosedhigher-level description should satisfy. Consequently, the application of information-theoretic measures of closure to ABMs allows us to quantify the information that is lost in the transition from the micro dynamics to a particular macro description. The method informs us in this way about the complexity of a system introduced by nontrivial interaction relations. Namely, if a favored level of observation is not compatible with the symmetries in !, a certain amount of memory is introduced by the transition from the micro level to such a macro description,and this is the fingerprintofemergenceinABMs.TheresultingdivergencefromMarkovianitycan be quantified using informationtheory,and the bookpresentsa scenario in which differentclosuremeasurescanbeexplicitlycomputed. Throughoutthe book, we mainly rely on two simple models to illustrate these theoreticalideas:thevotermodel(VM)andanextensionofitcalledthecontrarian voter model (CVM). Using these examples, the book shows that Markov chain theory allows for a rather precise understanding of the model dynamics in case of “simple” population structures where a tractable macro chain can be derived. Constrainingthesystembyinteractionnetworkswithastronglocalstructureleads to the emergence of meta-stable states in the transient of the model. Constraints onthe interactionbehaviorsuchas boundedconfidenceorassortativematinglead to the emergence of new absorbing states in the associated macro chain and are relatedtostablepatternsofpolarization(stablecoexistenceofdifferentopinionsor species). Constraints and heterogeneities in the microscopic system and complex social interactions are the basic characteristics of ABMs, and the Markov chain approach to link the micro chain to a macro-level description (and likewise the failureofaMarkovianlink)highlightsthecrucialroleplayedbythoseingredients inthegenerationofcomplexmacroscopicoutcomes. This book has developed out of my dissertation project at the department of Mathematical Physics at the University of Bielefeld. I am very grateful to my supervisorPhilippe Blanchard and to Dima Volchenkov(bothin Bielefeld) for an Preface ix open ear whenever I knocked on their doors and for the free environment they provided.IamalsoverygratefultoRicardoLima(Marseilles),whoisprobablythe personwhoengagedmostinthedetailsofthisproject,andtoTanyaAraújo(Lisbon) for her advice and encouragement. Chapter 8 has been developed in cooperation with Tanya. Especially the parts dealing with the application of information- theoreticmeasureshavebenefitedalotfromdiscussionswithEckehardOlbrichand Robin Lamarche-Perrin(both in Leipzig). All of this would have been a lot more difficultwithouttheunconditionalsupportofmyfamily. Finally,IgratefullyacknowledgefinancialsupportoftheGermanFederalMin- istry of Educationand Research (BMBF) throughthe projectLinguistic Networks (http://project.linguistic-networks.net) and the European Community’s Seventh FrameworkProgramme(FP7/2007-2013)undergrantagreementno.318723(Math- eMACS http://www.mathemacs.eu). Both projects have provided a very inspiring environment. Financial support by the Klaus Tschira Foundation (http://www. klaus-tschira-stiftung.de) during the finalization of this book is also gratefully acknowledged. Leipzig,Germany SvenBanisch
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