Relevant elments, Magnetization and Dynamical Properties in Kauffman Networks: a Numerical Study

TL;DR

This paper investigates the structure of Kauffman networks by numerically analyzing relevant elements, revealing their scaling behavior across different phases and providing insights into attractor properties in critical networks.

Contribution

It introduces a numerical study of relevant elements in Kauffman networks, confirming scaling predictions and enhancing understanding of attractor dynamics in critical regimes.

Findings

Number of relevant elements scales as sqrt(N) at criticality

Number of relevant elements scales linearly with N in the chaotic phase

Number of relevant elements is independent of system size in the frozen phase

Abstract

This is the first of two papers about the structure of Kauffman networks. In this paper we define the relevant elements of random networks of automata, following previous work by Flyvbjerg and Flyvbjerg and Kjaer, and we study numerically their probability distribution in the chaotic phase and on the critical line of the model. A simple approximate argument predicts that their number scales as sqrt(N) on the critical line, while it is linear with N in the chaotic phase and independent of system size in the frozen phase. This argument is confirmed by numerical results. The study of the relevant elements gives useful information about the properties of the attractors in critical networks, where the pictures coming from either approximate computation methods or from simulations are not very clear.