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Boolean dynamics of Kauffman models with a scale-free network.

Kazumoto Iguchi1, Shu-Ichi Kinoshita, Hiroaki S Yamada

  • 1KazumotoIguchi Research Laboratory, 70-3 Shinhari, Hari, Anan, Tokushima 774-0003, Japan.

Journal of Theoretical Biology
|April 6, 2007
PubMed
Summary

We investigated Boolean dynamics in quenched Kauffman models on scale-free networks. Results show state cycle lengths grow exponentially with network size, supporting a transition at k=2.

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Area of Science:

  • Complex Systems
  • Network Science
  • Theoretical Computer Science

Background:

  • Kauffman models are used to study gene regulatory networks.
  • Boolean dynamics in networks can exhibit complex behaviors like state cycles.
  • Network topology significantly influences system dynamics.

Purpose of the Study:

  • To investigate the Boolean dynamics of quenched Kauffman models on directed scale-free networks.
  • To compare these dynamics with those on directed random and exponential-fluctuation networks.
  • To analyze the impact of network size and average degree on state cycle length distributions.

Main Methods:

  • Numerical investigation of quenched Kauffman models.
  • Analysis of state cycle length distributions.
  • Comparison across different network topologies (scale-free, random, exponential-fluctuation).
  • Examination of system behavior with varying network size (N) and average degree (k).

Main Results:

  • In small networks (N≈150), state cycle length distributions (median, mean, std dev) grow exponentially with N for scale-free and exponential-fluctuation networks at k=2.
  • For larger networks (N≈10³), the median growth of attractor lengths transitions from algebraic to exponential as k approaches 2.
  • Observed growth patterns support the predicted transition at k(c)=2 from annealed models.

Conclusions:

  • The study provides numerical evidence for the transition in Boolean dynamics of quenched Kauffman models at k=2.
  • Scale-free and exponential-fluctuation network structures exhibit distinct behaviors compared to random networks.
  • Findings contribute to understanding the relationship between network topology and dynamical transitions in complex systems.