Finite size corrections to random Boolean networks

TL;DR

This paper investigates finite size effects in Boolean networks, comparing heuristic and exhaustive methods, and discusses the validity of analytical approximations for realistic network sizes.

Contribution

It provides a numerical comparison of belief propagation and exhaustive enumeration for finite networks and analyzes the annealed approximation's applicability.

Findings

Belief propagation closely matches exhaustive enumeration results for certain network sizes.

Finite size corrections significantly affect the predicted behavior of Boolean networks.

The annealed approximation's validity varies depending on network parameters.

Abstract

Since their introduction, Boolean networks have been traditionally studied in view of their rich dynamical behavior under different update protocols and for their qualitative analogy with cell regulatory networks. More recently, tools borrowed from statistical physics of disordered systems and from computer science have provided a more complete characterization of their equilibrium behavior. However, the largest part of the results have been obtained in the thermodynamic limit, which is often far from being reached when dealing with realistic instances of the problem. The numerical analysis presented here aims at comparing - for a specific family of models - the outcomes given by the heuristic belief propagation algorithm with those given by exhaustive enumeration. In the second part of the paper some analytical considerations on the validity of the annealed approximation are discussed.