Agent based models and opinion dynamics as Markov chains

This paper introduces a Markov chain approach that allows a rigorous analysis of agent based opinion dynamics as well as other related agent based models (ABM). By viewing the ABM dynamics as a micro description of the process, we show how the corresponding macro description is obtained by a projection construction. Then, well known conditions for lumpability make it possible to establish the cases where the macro model is stillMarkov. In this case we obtain a complete picture of the dynamics including the transient stage, the most interesting phase in applications. For such a purpose a crucial role is played by the type of probability distribution used to implement the stochastic part of the model which defines the updating rule and governs the dynamics. In addition, we show how restrictions in communication leading to the co–existence of different opinions follow from the emergence of new absorbing states. We describe our analysis in detail with some specific models of opinion dynamics. Generalizations concerning different opinion representations as well as opinion models with other interaction mechanisms are also discussed. We find that our method may be an attractive alternative to mean–field approaches and that this approach provides new perspectives on the modeling of opinion exchange dynamics, and more generally of other ABM.

[1]  Marsili,et al.  Nonequilibrium phase transition in a model for social influence , 2000, Physical review letters.

[2]  Jacob L. Moreno,et al.  Who shall survive? : foundations of sociometry, group psychotherapy, and sociodrama , 1953 .

[3]  Krapivsky,et al.  Exact results for kinetics of catalytic reactions. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[4]  Ehrhard Behrends.  Introduction to Markov Chains With Special Emphasis on Rapid Mixing , 2013 .

[5]  S. Redner,et al.  Voter model on heterogeneous graphs. , 2004, Physical review letters.

[6]  J. A. Barnes.  Class and Committees in a Norwegian Island Parish , 1954 .

[7]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[8]  F. Schweitzer,et al.  Nonlinear voter models: the transition from invasion to coexistence , 2003, cond-mat/0307742.

[9]  Joshua M. Epstein,et al.  Growing Artificial Societies: Social Science from the Bottom Up , 1996 .

[10]  S. Lukes.  Essays in Social Theory , 1978 .

[11]  Sebastian Grauwin,et al.  Dynamic Models of Residential Ségrégation: An Analytical Solution , 2010 .

[12]  A. Kellerman,et al.  The Constitution of Society : Outline of the Theory of Structuration , 2015 .

[13]  H. Garfinkel.  Studies in Ethnomethodology , 1968 .

[14]  F. Slanina,et al.  Analytical results for the Sznajd model of opinion formation , 2003, cond-mat/0305102.

[15]  S. Redner,et al.  Dynamics of majority rule in two-state interacting spin systems. , 2003, Physical review letters.

[16]  Guillaume Deffuant,et al.  Mixing beliefs among interacting agents , 2000, Adv. Complex Syst..

[17]  V. Climenhaga.  Markov chains and mixing times , 2013 .

[18]  F. Schweitzer.  Meinungsbildung , Kommunikation und Kooperation aus physikalischer Perspektive , 2003 .

[19]  R. Cont.  Volatility Clustering in Financial Markets: Empirical Facts and Agent-Based Models. , 2005 .

[20]  J. Moreno,et al.  Sociogram and Sociomatrix: A Note to the Paper by Forsyth and Katz , 1946 .

[22]  E. Rasmusen.  Private Truths, Public Lies: The Social Consequences of Preference Falsification , 1998 .

[23]  D. Helbing.  A Mathematical Model for the Behavior of Individuals in a Social Field , 1994, cond-mat/9805194.

[24]  P. Nielsen.  Optimal bonus strategies in life insurance: The Markov chain interest rate case , 2005 .

[25]  G. Cole.  Essays in social theory , 1951 .

[26]  Samuel D. Conte,et al.  Elementary Numerical Analysis: An Algorithmic Approach , 1975 .

[27]  Mark S. Granovetter.  The Strength of Weak Ties , 1973, American Journal of Sociology.

[28]  Jeffrey Alexander,et al.  The Micro-macro link , 1987 .

[29]  F. Vazquez,et al.  Analytical solution of the voter model on uncorrelated networks , 2008, 0803.1686.

[30]  P. Clifford,et al.  A model for spatial conflict , 1973 .

[31]  Timur Kuran.  Private Truths, Public Lies: The Social Consequences of Preference Falsification , 1995 .

[32]  T. Liggett,et al.  Stochastic Interacting Systems: Contact, Voter and Exclusion Processes , 1999 .

[33]  J. T. Cox,et al.  Coalescing Random Walks and Voter Model Consensus Times on the Torus in $\mathbb{Z}^d$ , 1989 .

[34]  B. Nordstrom.  FINITE MARKOV CHAINS , 2005 .

[35]  M. Kimura,et al.  The Stepping Stone Model of Population Structure and the Decrease of Genetic Correlation with Distance. , 1964, Genetics.

[36]  Wim Schoutens,et al.  Completion of a Lévy market by power-jump assets , 2005, Finance Stochastics.

[37]  Thomas C. Schelling,et al.  Dynamic models of segregation , 1971 .

[38]  Mark Pollicott,et al.  The Dynamics of Schelling-Type Segregation Models and a Nonlinear Graph Laplacian Variational Problem , 2001, Adv. Appl. Math..

[39]  R. Axelrod.  The Dissemination of Culture , 1997 .

[40]  Joshua M. Epstein,et al.  Growing Artificial Societies: Social Science from the Bottom Up , 1996 .

[41]  S. Fortunato,et al.  Statistical physics of social dynamics , 2007, 0710.3256.

[42]  Blake LeBaron,et al.  Agent-based computational finance : Suggested readings and early research , 2000 .

[43]  José Manuel Galán,et al.  Techniques to Understand Computer Simulations: Markov Chain Analysis , 2009, J. Artif. Soc. Soc. Simul..

[44]  J.-R. Chazottes,et al.  Projection of Markov Measures May Be Gibbsian , 2002 .

[45]  Rainer Hegselmann,et al.  Opinion dynamics and bounded confidence: models, analysis and simulation , 2002, J. Artif. Soc. Soc. Simul..