Kauffman Boolean model in undirected scale free networks

TL;DR

This paper analyzes the phase transition in undirected scale-free Boolean networks, revealing that the frozen-to-chaotic transition occurs for specific gamma values and discussing implications for modeling real gene regulatory networks.

Contribution

It provides an analytical and numerical study of the critical line in scale-free Boolean networks, highlighting the transition behavior for different gamma values and finite-size effects.

Findings

Transition occurs for 3<γ<3.5 in infinite networks.

Finite networks show the transition shifting towards smaller γ.

Percolation phenomena explain the unattainability of the critical line in simulations.

Abstract

We investigate analytically and numerically the critical line in undirected random Boolean networks with arbitrary degree distributions, including scale-free topology of connections $P(k)\sim k^{-\gamma}$. We show that in infinite scale-free networks the transition between frozen and chaotic phase occurs for $3<\gamma < 3.5$. The observation is interesting for two reasons. First, since most of critical phenomena in scale-free networks reveal their non-trivial character for $\gamma<3$, the position of the critical line in Kauffman model seems to be an important exception from the rule. Second, since gene regulatory networks are characterized by scale-free topology with $\gamma<3$, the observation that in finite-size networks the mentioned transition moves towards smaller $\gamma$ is an argument for Kauffman model as a good starting point to model real systems. We also explain that the unattainability of the critical line in numerical simulations of classical random graphs is due to percolation phenomena.