Exact dynamics of the critical Kauffman model with connectivity one

TL;DR

This paper provides an exact analysis of the critical Kauffman model with connectivity one, revealing that the number of attractors scales exponentially with the number of nodes in loops, demonstrating maximal complexity.

Contribution

It introduces a formalism for analyzing the dynamics of multiple loops and proves the exponential scaling of attractors, advancing understanding of critical Boolean networks.

Findings

Number of attractors scales as 2^m, where m is the number of nodes in loops.

Attractor complexity grows exponentially, faster than previously believed.

Provides a formalism for expressing dynamics of multiple loops.

Abstract

The critical Kauffman model with connectivity one is the simplest class of critical Boolean networks. Nevertheless, it exhibits intricate behavior at the boundary of order and chaos. We introduce a formalism for expressing the dynamics of multiple loops as a product of the dynamics of individual loops. Using it, we prove that the number of attractors scales as $2^m$, where $m$ is the number of nodes in loops - as fast as possible, and much faster than previously believed.