### Supercriticality of an annealed approximation of Boolean networks

**Daniel Valesin**

*(University of British Columbia)*

**Thomas Mountford**

*(École Polytechnique Fédérale de Lausanne)*

#### Abstract

We consider a model recently proposed by Chatterjee and Durrett as an "annealed approximation'' of boolean networks, which are a class of cellular automata on a random graph, as defined by S. Kauffman. The starting point is a random directed graph on $n$ vertices; each vertex has $r$ input vertices pointing to it. For the model of Chatterjee and Durrett, 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 Chatterjee and Durrett, we show that if $qr > 1$, then the time of persistence of activity of the dynamics is exponential in $n$.

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Pages: 1-12

Publication Date: May 4, 2013

DOI: 10.1214/ECP.v18-2479

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