The behavior of noise-resilient Boolean networks with diverse topologies

TL;DR

This paper analyzes how noise affects Boolean networks with various topologies, identifying phase transitions between memory-preserving and memoryless regimes, and providing bounds on noise resilience.

Contribution

It introduces a comprehensive analysis of noise resilience in Boolean networks with diverse topologies using stochastic blockmodels, including phase transition characterization.

Findings

Identifies a phase transition from non-ergodic to ergodic regimes.

Provides analytical expressions for average error and critical noise levels.

Establishes upper bounds on noise resilience for sparse networks.

Abstract

The dynamics of noise-resilient Boolean networks with majority functions and diverse topologies is investigated. A wide class of possible topological configurations is parametrized as a stochastic blockmodel. For this class of networks, the dynamics always undergoes a phase transition from a non-ergodic regime, where a memory of its past states is preserved, to an ergodic regime, where no such memory exists and every microstate is equally probable. Both the average error on the network, as well as the critical value of noise where the transition occurs are investigated analytically, and compared to numerical simulations. The results for "partially dense" networks, comprised of relatively few, but dynamically important nodes, which have a number of inputs which greatly exceeds the average for the entire network, give very general upper bounds on the maximum resilience against noise attainable on globally sparse systems.