Relevant components in critical random Boolean networks

TL;DR

This paper analyzes the structure of relevant components in critical random Boolean networks, revealing how their number, size, and complexity depend on network parameters through analytical and numerical methods.

Contribution

It provides a detailed analysis of relevant components in critical random Boolean networks, including their distribution, size, and topology, independent of update schemes.

Findings

Number of relevant components grows logarithmically with network size

Most relevant nodes with multiple relevant inputs are in the same component

Distribution of complex components becomes size-independent in large networks

Abstract

Random Boolean networks were introduced in 1969 by Kauffman as a model for gene regulation. By combining analytical arguments and efficient numerical simulations, we evaluate the properties of relevant components of critical random Boolean networks independently of update scheme. As known from previous work, the number of relevant components grows logarithmically with network size. We find that in most networks all relevant nodes with more than one relevant input sit in the same component, while all other relevant components are simple loops. As the proportion of nonfrozen nodes with two relevant inputs increases, the number of relevant components decreases and the size and complexity of the largest complex component grows. We evaluate the probability distribution of different types of complex components in an ensemble of networks and confirm that it becomes independent of network size in the limit of large network size. In this limit, we determine analytically the frequencies of occurence of complex components with different topologies.