Noise in random Boolean networks

TL;DR

This paper studies how noise affects the dynamics of Random Boolean Networks, revealing different transition behaviors depending on the number of connections per node, with analytical and simulation results characterizing these effects.

Contribution

It provides a comprehensive analysis of noise effects on RBNs, including new order parameters and analytical methods for their evaluation across different connectivity regimes.

Findings

Smooth transition from deterministic to stochastic dynamics for K≤2.

First order transition at p=0 for K>2.

Distribution of frozenness exhibits fractal structure for K>1.

Abstract

We investigate the effect of noise on Random Boolean Networks. Noise is implemented as a probability $p$ that a node does not obey its deterministic update rule. We define two order parameters, the long-time average of the Hamming distance between a network with and without noise, and the average frozenness, which is a measure of the extent to which a node prefers one of the two Boolean states. We evaluate both order parameters as function of the noise strength, finding a smooth transition from deterministic ($p=0$) to fully stochastic ($p=1/2$) dynamics for networks with $K\le2$, and a first order transition at $p=0$ for $K>2$. Most of the results obtained by computer simulation are also derived analytically. The average Hamming distance can be evaluated using the annealed approximation. In order to obtain the distribution of frozenness as function of the noise strength, more sophisticated self-consistent calculations had to be performed. This distribution is a collection of delta peaks for K=1, and it has a fractal sructure for $K>1$, approaching a continuous distribution in the limit $K\gg1$.