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Scaling in critical random Boolean networks.

Viktor Kaufman1, Tamara Mihaljev, Barbara Drossel

  • 1Institut für Festkörperphysik, TU Darmstadt, Hochschulstrasse 6, 64289 Darmstadt, Germany.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 31, 2005
PubMed
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We analyzed critical Kauffman networks and found that the number of nonfrozen nodes scales as N(2/3) and relevant nodes as N(1/3). Most relevant components are simple loops, with attractor size growing exponentially.

Area of Science:

  • Complex systems
  • Statistical physics
  • Network theory

Background:

  • Kauffman networks are models of gene regulatory networks.
  • Understanding the collective behavior of such networks is crucial for systems biology.
  • Previous studies have explored phase transitions in these networks.

Purpose of the Study:

  • To analytically derive the scaling behavior of nonfrozen and relevant nodes in critical Kauffman networks.
  • To characterize the structure of the frozen core and relevant components.
  • To determine the scaling of attractors in these networks.

Main Methods:

  • Analytical derivation using stochastic processes.
  • Analysis in the thermodynamic limit (large network size N).
  • Characterization of node states (frozen vs. nonfrozen, relevant vs. irrelevant).

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Main Results:

  • The mean number of nonfrozen nodes scales as N(2/3) in critical Kauffman networks.
  • A subset of nonfrozen nodes, scaling as N(1/3), possess two nonfrozen inputs.
  • The mean number of relevant nodes scales as N(1/3), with most relevant components being simple loops.

Conclusions:

  • The structure of critical Kauffman networks is characterized by a dominant nonfrozen core and a sparse set of relevant nodes.
  • Attractor size and number increase faster than any power law with network size.
  • These findings provide insights into the organization and dynamics of complex biological networks.