Abstract
Boolean networks can be used as simple but general models for complex
self-organizing systems. The freedom to choose different rules and
structures of interactions makes this model applicable to a wide variety
of complex phenomena. It is known that the damage dynamics in annealed
Boolean systems should fall in the same universality class of the
directed percolation model. In this work we present results about the
behavior of this model at and near the critically ordered condition for
both the annealed and the quenched versions of the model. Our study
concentrates on the way the system responds to a small perturbation. We
show that the characteristic correlation time, i.e., the time in which
any memory of this perturbation is lost, diverges as one moves towards
criticality. Exactly at the critical point, we observe that the time for
returning to the natural state after the perturbation follows a
power-law distribution. This indicates that most perturbations are
quickly restored, while few events may have a global effect on the
system, suggesting a mechanism that assures at the same time robustness
and adaptability. The critical exponents obtained are in agreement with
the values expected for the universality class of mean-field directed
percolation both in the annealed and in the quenched Boolean network
model. This gives further evidence that annealed Boolean networks may in
certain conditions provide a good model for understanding the behavior
of regulatory systems. Our results may give insight into the way real
self-organizing systems respond to external stimuli, and why critically
ordered systems are often observed in Nature. (C) 2008 Published by
Elsevier B.V.
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