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Difference equation for tracking perturbations in systems of Boolean nested canalyzing functions.

Elena S Dimitrova1, Oleg I Yordanov2, Mihaela T Matache3

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Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
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This study models how perturbations fade in networks using canalyzing Boolean functions. Perturbation effects diminish over time in networks with sufficiently canalyzing functions, revealing percolation limits.

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Area of Science:

  • Systems Biology
  • Network Science
  • Computational Biology

Background:

  • Boolean functions are fundamental to modeling complex systems, including gene regulatory networks.
  • Canalyzing properties in Boolean functions, where one input fixes the output, simplify network behavior analysis.
  • Partially nested canalyzing functions offer a realistic model for cascading influences in biological and other networks.

Purpose of the Study:

  • To develop and analyze a mathematical model for the spread of perturbations in networks of partially nested canalyzing Boolean functions.
  • To investigate the temporal dynamics of perturbation effects and identify conditions under which they decay.
  • To explore the relationship between perturbation impact, network sensitivity, and percolation phenomena.

Main Methods:

  • Development of a difference equation model to track the probability of a node being affected by a perturbation over time.
  • Numerical validation of the model to assess its accuracy in simulating perturbation spread.
  • Analysis of network parameters, including average sensitivity and the degree of canalyzation, to understand their influence on perturbation dynamics.

Main Results:

  • Demonstrated that perturbations decay to zero over time in networks with sufficiently canalyzing Boolean functions.
  • Quantified the maximum dynamical impact of a perturbation, showing it is comparable to the average impact across various network sensitivities.
  • Identified percolation limits, critical parameter values where the expected perturbation effect transitions to zero.

Conclusions:

  • Networks employing canalyzing Boolean functions exhibit inherent stability against the spread of perturbations.
  • The study provides insights into the robustness of complex networks and the critical role of function properties in determining system dynamics.
  • Understanding percolation limits is crucial for predicting network behavior and designing resilient systems.