Phase transitions and memory effects in the dynamics of Boolean networks

TL;DR

This paper employs the generating functional method to analyze the dynamics of Boolean networks, revealing phase transitions, memory effects, and the validity of approximations across various topologies and noise levels.

Contribution

It introduces an exact analytical framework for Boolean network dynamics that accounts for topology, Boolean functions, noise, and memory effects, surpassing previous approximation methods.

Findings

Identifies conditions for the validity of the annealed approximation.

Explores phase transitions under different noise levels.

Analyzes models with strong memory effects where approximations fail.

Abstract

The generating functional method is employed to investigate the synchronous dynamics of Boolean networks, providing an exact result for the system dynamics via a set of macroscopic order parameters. The topology of the networks studied and its constituent Boolean functions represent the system's quenched disorder and are sampled from a given distribution. The framework accommodates a variety of topologies and Boolean function distributions and can be used to study both the noisy and noiseless regimes; it enables one to calculate correlation functions at different times that are inaccessible via commonly used approximations. It is also used to determine conditions for the annealed approximation to be valid, explore phases of the system under different levels of noise and obtain results for models with strong memory effects, where existing approximations break down. Links between BN and general Boolean formulas are identified and common results to both system types are highlighted.