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Stochastic Analysis of Cellular Automata and the Voter Model

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Cellular Automata (ACRI 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2493))

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Abstract

We make a stochastic analysis of both deterministic and stochastic cellular automata. The theory uses a mesoscopic view, i.e. it works with probabilities instead of individual configurations used in micro-simulations. We make an exact analysis by using the theory of Markov processes. This can be done for small problems only. For larger problems we approximate the distribution by products of marginal distributions of low order. The approximation use new developments in efficient computation of probabilities based on factorizations of the distribution. We investigate the popular voter model. We show that for one dimension the bifurcation at α = 1/3 is an artifact of the mean-field approximation.

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References

  1. U. N. Bhat Elements of Applied Stochastic Processes. Wiley, New York, 1984.

    MATH  Google Scholar 

  2. U. Diekmann, R. Law, and J. Metz (eds.). The Geometry of Ecological Interactions. Cambridge University Press, Cambridge, 2000.

    Google Scholar 

  3. R. Durret. http://www.math.cornell.edu/~durrett/survey/survhome.html

  4. H.A. Gutowitz and J.D. Victor and B.W. Knight. Local structure theory for cellular automata. Physica D, 28D:18–48, 1987.

    Article  MathSciNet  Google Scholar 

  5. H.A. Gutowitz and J.D. Victor. Local structure theory in more than one dimension. Complex Systems, 1:57–67, 1987.

    MATH  MathSciNet  Google Scholar 

  6. D. R Heyman and D. P. OLeary. Overcoming instability in computing the fundamental matrix for a Markov chain. SIAM J. Matrix Analysis and Applications, 19:534–540, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  7. A. Ilachinski Cellular Automata A Discrete Universe. World Scientific, Singapore, 2001

    MATH  Google Scholar 

  8. M.I. Jordan. Learning in Graphical Models. MIT Press, Cambrigde, 1999

    Google Scholar 

  9. S. L. Lauritzen. Graphical Models. Oxford: Clarendom Press, 1996.

    Google Scholar 

  10. H. Muhlenbein. Darwin’s continent cycle theory and its simulation by the Prisoner’s Dilemma. Complex Systems, 5:459–478, 1991.

    MathSciNet  Google Scholar 

  11. Heinz Muhlenbein, Thilo Mahnig, and Alberto Ochoa. Schemata, distributions and graphical models in evolutionary optimization. Journal of Heuristics, 5(2):213–247, 1999.

    Article  Google Scholar 

  12. H. Muhlenbein and Th. Mahnig. Evolutionary algorithms: From recombination to search distributions. In L. Kallel, B. Naudts, and A. Rogers, editors, Theoretical Aspects of Evolutionary Computing, Natural Computing, pages 137–176. Springer Verlag, Berlin, 2000.

    Google Scholar 

  13. Heinz Muhlenbein and Thilo Mahnig. Evolutionary optimization and the estimation of search distributions Journal of Approx. Reason., to appear, 2002.

    Google Scholar 

  14. N. A. Oomes. Emerging markets and persistent inequality in a nonlinear voter model. In D. Griffeath and C. Moore, editors, New Constructions in Cellular Automata, pages 207–229, Oxford University Press, Oxford, 2002.

    Google Scholar 

  15. M. Opper and D. Saad, editors. Advanced Mean Field Methods, MIT Press, Cambridge, 2001.

    MATH  Google Scholar 

  16. S. Wolfram. Cellular Automata and Complexity. Addison-Wesley, Reading, 1994.

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. FhG-AiS, D-53754, Sankt Augustin

    Heinz Mühlenbein & Robin Höns

Authors
  1. Heinz Mühlenbein
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  2. Robin Höns
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Systems and Communication, University of Milano-Bicocca, Via Bicocca degli Arcimboldi, 8, 20126, Milan, Italy

    Stefania Bandini

  2. Computer Science Department, University of Geneva, 24, Rue du General Dufour, 1211, Geneva 4, Switzerland

    Bastien Chopard

  3. Computer Science Institute Faculty of Sciences, University of Lausanne, 1015, Lausanne, Switzerland

    Marco Tomassini

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© 2002 Springer-Verlag Berlin Heidelberg

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Mühlenbein, H., Höns, R. (2002). Stochastic Analysis of Cellular Automata and the Voter Model. In: Bandini, S., Chopard, B., Tomassini, M. (eds) Cellular Automata. ACRI 2002. Lecture Notes in Computer Science, vol 2493. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45830-1_9

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  • DOI: https://doi.org/10.1007/3-540-45830-1_9

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-44304-9

  • Online ISBN: 978-3-540-45830-2

  • eBook Packages: Springer Book Archive

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