Supercriticality of Annealed Approximations of Boolean Networks

TL;DR

This paper analyzes an annealed approximation of Boolean networks modeled as a threshold contact process on a random graph, demonstrating that activity persists exponentially long when the product of parameters exceeds one.

Contribution

It improves previous results by proving exponential persistence time of activity when the product of parameters exceeds one in the model.

Findings

Persistence time of activity is exponential in network size n when qr > 1.

The model exhibits a phase transition at qr = 1.

The analysis extends understanding of dynamics in annealed Boolean network approximations.

Abstract

We consider a model recently proposed by Chatterjee and Durrett [CD2011] as an "annealed approximation" of boolean networks, which are a class of cellular automata on a random graph, as defined by S. Kauffman [K69]. The starting point is a random directed graph on $n$ vertices; each vertex has $r$ input vertices pointing to it. For the model of [CD2011], a discrete time threshold contact process is then considered on this graph: at each instant, each vertex has probability $q$ of choosing to receive input; if it does, and if at least one of its input vertices were in state 1 at the previous instant, then it is labelled with a 1; in all other cases, it is labelled with a 0. $r$ and $q$ are kept fixed and $n$ is taken to infinity. Improving a result of [CD2011], we show that if $qr > 1$, then the time of persistence of activity of the dynamics is exponential in $n$.