Power-Law Distributions in Some Random Boolean Networks

The Kauffman net is a dynamical system of logical variables receiving two random inputs and each randomly assigned a Boolean function. We show that the attractor and transient lengths exhibit scaleless behavior with power-law distributions over up to 10 orders of magnitude in probability. Our results provide evidence for the existence of the “edge of chaos” as a distinct regime between the ordered and chaotic phases analogous to a critical point in statistical mechanics. The power-law distributions are robust to the changes in the composition of the transition rules and network dynamics.