Dynamical Instability in Boolean Networks as a Percolation Problem

Boolean networks, a widely used model of gene regulatory networks, exhibit a phase transition between a stable regime, in which small perturbations die out, and an unstable regime, in which small perturbations grow exponentially. We show that this phase transition can be mapped onto a static percolation problem which predicts the critical point and the long-time Hamming distance between perturbed and unperturbed systems. The results, which apply to Boolean networks with a broad class of topologies and update functions, are confirmed by numerical simulations.