Number and length of attractors in a critical Kauffman model with connectivity one

TL;DR

This paper proves that in critical Kauffman models with connectivity one, the average number and length of attractors grow faster than any power law as the network size increases, highlighting complex dynamics at criticality.

Contribution

It provides a rigorous proof that attractor number and length grow faster than any power law in critical connectivity-one Boolean networks, using a growth process and lower bounds.

Findings

Mean number of attractors increases faster than any power law.

Mean attractor length increases faster than any power law.

Results apply to critical random Boolean networks with connectivity one.

Abstract

The Kauffman model describes a system of randomly connected nodes with dynamics based on Boolean update functions. Though it is a simple model, it exhibits very complex behavior for "critical" parameter values at the boundary between a frozen and a disordered phase, and is therefore used for studies of real network problems. We prove here that the mean number and mean length of attractors in critical random Boolean networks with connectivity one both increase faster than any power law with network size. We derive these results by generating the networks through a growth process and by calculating lower bounds.