Cyclic automata networks on finite graphs

  • Martín Matamala
  • Eric Goles
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 911)


We study the principal dynamical aspects of the cyclic automata on finite graphs.

We give bounds in the transient time and periodicity depending essentially on the graph structure. It is important to point out that there exist non-polynomial periods \(e^\Omega (\sqrt {\left| V \right|} )\), where V denotes the number of sites in the graph.

To obtain these results we introduce some mathematical tool as continuity, firing paths, jump and efficiency, which are interesting by themselves because they give a strong mathematical framework to study such discrete dynamical systems.


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  1. 1.
    J. P. Allouche and Ch. Reder. Oscillations spatio-temporelles engendrées par un automate cellulaire. Disc. Appl. Maths, pages 215–254, 1984.Google Scholar
  2. 2.
    M. Bramson and D. Griffeath. Flux and fixation in cyclic particle system. Ann. Probability, 17(1):26–45, 1989.Google Scholar
  3. 3.
    Robert Fisch. Cyclic cellular automata and related processes. Physica D, 45:19–25, 1990.Google Scholar
  4. 4.
    E. Goles and S. Martínez. Neural and Automata Networks. Kluwer Pub., 1990.Google Scholar
  5. 5.
    J. M. Greenberg, B. D. Hassard and S.P. Hastings. Patterns formation and periodic structures in systems modeled by reaction-diffusion equations. Bull. Amer. Math. Soc., 34(3):515–523, 1978.Google Scholar
  6. 6.
    J. M. Greenberg, C. Greene and S. Hastings C. A combinatorial problem arising in the study of reaction-diffusion equations. SIAM J. on Alg. and Disc. Maths, 1(1):34–42, 1980.Google Scholar
  7. 7.
    J.M. Greenberg and S.P. Hasting. Spatial patterns for discrete models of diffusion in excitable media. SIAM Journal Appl. Maths., 34(3):515–523, 1978.Google Scholar
  8. 8.
    D. Griffeath. Excitable Cellular Automata. Proc of workshop on cellular automata(center for scientific computing, Espoo, Finland), 1991.Google Scholar
  9. 9.
    G. H. Hardy and E. M. Wright. An introduction to the Theory of Numbers Oxford University Press, New York, fifth edition, 1979.Google Scholar
  10. 10.
    R. Shingai. Maximum period on 2-dimensional uniform neural networks. Inf. and Control, 41:324–341, 1979.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Martín Matamala
    • 1
  • Eric Goles
    • 1
  1. 1.Departamento de Ingeniería MatemáticaU. de ChileSantiagoChile

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