On the number of attractors in random Boolean networks

TL;DR

This paper generalizes the analysis of attractor counts in random Boolean networks, linking theoretical calculations with network structure concepts, and explains how attractor numbers scale with system size.

Contribution

It extends previous work to critical networks with varying input counts and update function distributions, connecting mathematical terms to network components and providing a size-dependent attractor estimate.

Findings

Derived formulas for attractor numbers in generalized networks

Linked calculation terms to network structural features

Proposed a size-dependent attractor scaling argument

Abstract

The evaluation of the number of attractors in Kauffman networks by Samuelsson and Troein is generalized to critical networks with one input per node and to networks with two inputs per node and different probability distributions for update functions. A connection is made between the terms occurring in the calculation and between the more graphic concepts of frozen, nonfrozen and relevant nodes, and relevant components. Based on this understanding, a phenomenological argument is given that reproduces the dependence of the attractor numbers on system size.