Persistence of Activity in Threshold Contact Processes, an "Annealed Approximation" of Random Boolean Networks

TL;DR

This paper models gene regulatory networks using a threshold contact process on random graphs, showing conditions under which the network exhibits persistent, chaotic activity, with persistence times growing exponentially with network size.

Contribution

It introduces an annealed approximation for random Boolean networks and establishes conditions for long-term activity persistence in the model.

Findings

Persistence occurs when r·2p(1-p)>1.

Persistence time grows at least exponentially with network size.

Chaotic behavior is linked to specific parameter thresholds.

Abstract

We consider a model for gene regulatory networks that is a modification of Kauffmann's (1969) random Boolean networks. There are three parameters: $n =$ the number of nodes, $r =$ the number of inputs to each node, and $p =$ the expected fraction of 1's in the Boolean functions at each node. Following a standard practice in the physics literature, we use a threshold contact process on a random graph on $n$ nodes, in which each node has in degree $r$, to approximate its dynamics. We show that if $r\ge 3$ and $r \cdot 2p(1-p)>1$, then the threshold contact process persists for a long time, which correspond to chaotic behavior of the Boolean network. Unfortunately, we are only able to prove the persistence time is $\ge \exp(cn^{b(p)})$ with $b(p)>0$ when $r\cdot 2p(1-p)> 1$, and $b(p)=1$ when $(r-1)\cdot 2p(1-p)>1$.