Scaling in critical random Boolean networks

TL;DR

This paper analytically derives the scaling laws for nonfrozen and relevant nodes in critical random Boolean networks, revealing how these quantities grow with network size and their structural implications.

Contribution

It introduces stochastic processes to analytically determine the scaling behavior of nonfrozen and relevant nodes in critical Kauffman networks, providing new insights into their structure.

Findings

Number of nonfrozen nodes scales as N^{2/3}

Number of relevant nodes scales as N^{1/3}

Attractors grow faster than any power law with network size

Abstract

We derive mostly analytically the scaling behavior of the number of nonfrozen and relevant nodes in critical Kauffman networks (with two inputs per node) in the thermodynamic limit. By defining and analyzing a stochastic process that determines the frozen core we can prove that the mean number of nonfrozen nodes scales with the network size N as N^{2/3}, with only N^{1/3} nonfrozen nodes having two nonfrozen inputs. We also show the probability distributions for the numbers of these nodes. Using a different stochastic process, we determine the scaling behavior of the number of relevant nodes. Their mean number increases for large N as N^{1/3}, and only a finite number of relevant nodes have two relevant inputs. It follows that all relevant components apart from a finite number are simple loops, and that the mean number and length of attractors increases faster than any power law with network size.