Electronic Communications in Probability
- Electron. Commun. Probab.
- Volume 18 (2013), paper no. 32, 12 pp.
Supercriticality of an annealed approximation of Boolean networks
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 $n$
Electron. Commun. Probab., Volume 18 (2013), paper no. 32, 12 pp.
Accepted: 4 May 2013
First available in Project Euclid: 7 June 2016
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Primary: Interacting Particle Systems
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Valesin, Daniel; Mountford, Thomas. Supercriticality of an annealed approximation of Boolean networks. Electron. Commun. Probab. 18 (2013), paper no. 32, 12 pp. doi:10.1214/ECP.v18-2479. https://projecteuclid.org/euclid.ecp/1465315571