Fixed points of Boolean networks with sparse connections

TL;DR

This paper analyzes fixed points in Boolean networks on sparse random graphs, revealing phase transitions, clustering of fixed points, and singularities in their statistical moments.

Contribution

It provides a detailed characterization of fixed points, their organization into clusters, and the nature of phase transitions in sparse Boolean networks.

Findings

Moments of the number of fixed points remain finite except at phase transitions.

Fixed points form clusters with properties depending on the phase.

Distribution of fixed points is fully characterized in the frozen phase.

Abstract

We study fixed points of cellular automata with $N$ sites on random sparse graphs. In the large $N$ limit such models are known to exhibit phase transitions, from a ``frozen'' phase, where at most a finite number of sites fluctuate at long times, to a ``fluctuating'' phase where a finite fraction of sites fluctuate. We consider several models, calculating the first and second moments of the number of fixed points, and find that these moments remain finite in the large $N$ limit, except at the transitions where they become singular. The singularities can take several forms, including divergence of the mean or variance of the number of fixed points, on one or both sides of the transition. The type of singularity is related to properties of the mean field dynamics or dynamics of the distance between copies of the system. In configuration space, we find that fixed points are organized into clusters, each consisting of sets of fixed points that agree with one another except for on a finite number of sites. In the frozen phase there is only one cluster, while in the fluctuating phase there may be multiple clusters. If there are multiple clusters, the distance between fixed points in different clusters is extensive. We show that the differences within the clusters correspond to local changes near short cycles in the directed graph of connections whose influence is eventually limited. In the frozen phase, we calculate the full distribution of the number of fixed points.