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Random maps and attractors in random Boolean networks.

Björn Samuelsson1, Carl Troein

  • 1Complex Systems Division, Department of Theoretical Physics, Lund University, Sölvegatan 14A, S-223 62 Lund, Sweden. bjorn@thep.lu.se

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 31, 2005
PubMed
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This study analyzes random Boolean networks using random map theory to find analytical results for attractors. The findings offer insights into network dynamics and highlight challenges in comparing models to real systems.

Area of Science:

  • Complex Systems
  • Network Theory
  • Dynamical Systems

Background:

  • Random Boolean networks (RBNs) exhibit complex dynamics despite simple rules.
  • Attractors and their properties are key to understanding RBN behavior.
  • The topology of RBNs with one input per node can be modeled as random maps.

Purpose of the Study:

  • To develop an analytical approach for studying attractors in RBNs using random map theory.
  • To apply these analytical results to non-chaotic networks.
  • To investigate differences between average and typical attractor numbers in RBNs.

Main Methods:

  • Modeling RBNs with one input per node as random maps.
  • Deriving analytical results for attractors based on random map topology.

Related Experiment Videos

  • Approximating non-chaotic networks using the random map framework.
  • Analyzing observables like the number of components and cumulants in random maps.

    • Main Results:

    • Achieved good agreement when approximating non-chaotic networks with the random map model, particularly for attractor numbers.
    • Identified significant differences between average and typical attractor numbers, underscoring challenges in direct RBN-system comparisons.
    • Derived new results for random maps, including component distribution and asymptotic expansions for cumulants.

    Conclusions:

    • The random map approach provides a powerful analytical tool for understanding attractors in specific types of RBNs.
    • Direct comparison of RBNs with real systems requires careful consideration of attractor number distributions.
    • The study advances the mathematical understanding of random maps and their applications to network dynamics.