On the properties of cycles of simple Boolean networks

TL;DR

This paper analytically investigates the cycle properties of simple Boolean networks, revealing that their cycle numbers and lengths can diverge faster than any power law, providing insights into critical Kauffman networks.

Contribution

It offers a mostly analytical characterization of cycle counts and lengths in specific simple Boolean networks, enhancing understanding of their dynamics.

Findings

Mean number of cycles can diverge faster than any power law

Cycle lengths can grow extremely rapidly, exceeding polynomial growth

Features observed in Kauffman networks are confirmed in these simple models

Abstract

We study two types of simple Boolean networks, namely two loops with a cross-link and one loop with an additional internal link. Such networks occur as relevant components of critical K=2 Kauffman networks. We determine mostly analytically the numbers and lengths of cycles of these networks and find many of the features that have been observed in Kauffman networks. In particular, the mean number and length of cycles can diverge faster than any power law.