Dynamical Instability in Boolean Networks as a Percolation Problem

TL;DR

This paper demonstrates that the phase transition in Boolean network dynamics, relevant for gene regulation models, can be understood through a static percolation problem, predicting long-term behavior of perturbations.

Contribution

It introduces a novel mapping of Boolean network phase transitions onto a static percolation problem, providing a new analytical approach.

Findings

Percolation mapping accurately predicts phase transition points.

Long-term Hamming distance can be derived from static percolation properties.

Numerical verification confirms the theoretical predictions.

Abstract

Boolean networks, widely used to model gene regulation, exhibit a phase transition between regimes in which small perturbations either die out or grow exponentially. We show and numerically verify that this phase transition in the dynamics can be mapped onto a static percolation problem which predicts the long-time average Hamming distance between perturbed and unperturbed orbits.