Attractor period distribution for critical Boolean networks

TL;DR

This paper analytically demonstrates that in large critical Boolean networks, the distribution of attractor periods follows a power-law, supported by numerical sampling, highlighting the role of relevant components in network dynamics.

Contribution

The paper introduces an analytic approach using relevant components to derive the power-law distribution of attractor periods in critical Boolean networks.

Findings

Attractor periods follow a power-law distribution with exponent -1.

Numerical sampling supports the analytic predictions.

Relevant components determine the network's attractor dynamics.

Abstract

Using analytic arguments, we show that dynamical attractor periods in large critical Boolean networks are power-law distributed. Our arguments are based on the method of relevant components, which focuses on the behavior of the nodes that control the dynamics of the entire network and thus determine the attractors. Assuming that the attractor period is equal to the least common multiple of the size of all relevant components, we show that the distribution in large networks is well approximated by a power-law with an exponent of -1. Numerical evidence based on sampling of attractors supports the conclusions of our analytic arguments.