Power-law Distributions in the Kauffman Net

TL;DR

This paper demonstrates that Kauffman nets exhibit power-law distributions in attractor and transient lengths, supporting the concept of an 'edge of chaos' phase between order and chaos, with robust properties across various dynamics.

Contribution

It provides evidence that Kauffman nets have scale-free behavior in their dynamics, highlighting a critical phase akin to a phase transition in statistical mechanics.

Findings

Attractor and transient lengths follow power-law distributions over ten orders of magnitude.

The power-law behavior is robust to changes in transition rules and network dynamics.

Supports the existence of an 'edge of chaos' phase in Kauffman nets.

Abstract

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 ten orders of magnitude. Our results provide evidence for the existence of the "edge of chaos" as a distinct phase between the ordered and chaotic regimes 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.