Random maps and attractors in random Boolean networks

TL;DR

This paper introduces an analytical approach to studying attractors in random Boolean networks by modeling their topology as random maps, providing new insights into their dynamical behaviors and attractor distributions.

Contribution

The authors develop an analytical framework for random maps in Boolean networks, enabling precise predictions of attractor properties and extending results to related non-chaotic networks.

Findings

Good agreement between analytical results and numerical simulations of attractor counts

Derived distribution of the number of components in random maps

Asymptotic expansions for cumulants up to the 4th order

Abstract

Despite their apparent simplicity, random Boolean networks display a rich variety of dynamical behaviors. Much work has been focused on the properties and abundance of attractors. The topologies of random Boolean networks with one input per node can be seen as graphs of random maps. We introduce an approach to investigating random maps and finding analytical results for attractors in random Boolean networks with the corresponding topology. Approximating some other non-chaotic networks to be of this class, we apply the analytic results to them. For this approximation, we observe a strikingly good agreement on the numbers of attractors of various lengths. We also investigate observables related to the average number of attractors in relation to the typical number of attractors. Here, we find strong differences that highlight the difficulties in making direct comparisons between random Boolean networks and real systems. Furthermore, we demonstrate the power of our approach by deriving some results for random maps. These results include the distribution of the number of components in random maps, along with asymptotic expansions for cumulants up to the 4th order.