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Dynamical behavior and influence of stochastic noise on certain generalized Boolean networks

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  • Beck, Gary L.
  • Matache, Mihaela T.

Abstract

This study considers a simple Boolean network with N nodes, each node’s state at time t being determined by a certain number of parent nodes. The network is analyzed when the connectivity k is fixed or variable. Making use of a Boolean rule that is a generalization of Rule 22 of elementary cellular automata, a generalized formula for providing the probability of finding a node in state 1 at time t is determined. We show typical behaviors of the iterations, and we study the dynamics of the network through Lyapunov exponents, bifurcation diagrams and fixed point analysis. We conclude that the network may exhibit stability or chaos, depending on the underlying parameters. In general high connectivity is associated with a convergence to zero of the probability of finding a node in state 1 at time t. We also study analytically and numerically the dynamics of the network under a stochastic noise procedure. We show that under a smaller probability of disturbing the nodes through the noise procedure the system tends to exhibit more nodes in the same state. For many parameter combinations there is no critical value of the noise parameter below which the network remains organized and above which it behaves randomly.

Suggested Citation

  • Beck, Gary L. & Matache, Mihaela T., 2008. "Dynamical behavior and influence of stochastic noise on certain generalized Boolean networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(19), pages 4947-4958.
    • Handle: RePEc:eee:phsmap:v:387:y:2008:i:19:p:4947-4958
    DOI: 10.1016/j.physa.2008.04.032
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    References listed on IDEAS

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    1. Goodrich, Christopher S. & Matache, Mihaela T., 2007. "The stabilizing effect of noise on the dynamics of a Boolean network," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 379(1), pages 334-356.
    2. Fan, Jin & Li, Xiang & Fan Wang, Xiao, 2005. "On synchronous preference of complex dynamical networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 355(2), pages 657-666.
    3. Marr, Carsten & Hütt, Marc-Thorsten, 2005. "Topology regulates pattern formation capacity of binary cellular automata on graphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 354(C), pages 641-662.
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    Cited by:

    1. Hung, Yao-Chen & Lin, Chai-Yu, 2014. "Modeling intrinsic noise in random Boolean networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 395(C), pages 121-127.

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