Scaling in ordered and critical random Boolean networks

TL;DR

This paper investigates the scaling behavior of relevant nodes and attractors in ordered and critical random Boolean networks, revealing constant and N^{1/3} growth patterns, respectively, and explaining the asymptotic scaling observed in large systems.

Contribution

It provides rigorous analysis and numerical evidence for the scaling laws of relevant nodes and attractors in ordered and critical regimes of random Boolean networks.

Findings

Relevant nodes remain constant in ordered networks as size increases.

Number of relevant nodes scales as N^{1/3} in critical networks.

Median number of attractors grows faster than linearly with N.

Abstract

Random Boolean networks, originally invented as models of genetic regulatory networks, are simple models for a broad class of complex systems that show rich dynamical structures. From a biological perspective, the most interesting networks lie at or near a critical point in parameter space that divides ``ordered'' from ``chaotic'' attractor dynamics. In the ordered regime, we show rigorously that the average number of relevant nodes (the ones that determine the attractor dynamics) remains constant with increasing system size N. For critical networks, our analysis and numerical results show that the number of relevant nodes scales like N^{1/3}. Numerical experiments also show that the median number of attractors in critical networks grows faster than linearly with N. The calculations explain why the correct asymptotic scaling is observed only for very large N.