Dynamics of Boolean networks - an exact solution

TL;DR

This paper provides an exact analytical framework for understanding the dynamics of Boolean networks with quenched disorder and noise, revealing when common approximations are valid and offering new insights into their stationary states.

Contribution

It develops a general generating functional approach for Boolean networks, enabling exact solutions and analysis of their dynamics and stationary states, surpassing previous methods.

Findings

Exact solutions for Boolean network dynamics with disorder and noise

Identification of cases where annealed approximation is valid or invalid

Insights into stationary state properties and links to Boolean formulas

Abstract

The dynamics of Boolean networks (BN) with quenched disorder and thermal noise is studied via the generating functional method. A general formulation, suitable for BN with any distribution of Boolean functions, is developed. It provides exact solutions and insight into the evolution of order parameters and properties of the stationary states, which are inaccessible via existing methodology. We identify cases where the commonly used annealed approximation is valid and others where it breaks down. Broader links between BN and general Boolean formulas are highlighted.