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2013 Supercriticality of an annealed approximation of Boolean networks
Daniel Valesin, Thomas Mountford
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Electron. Commun. Probab. 18: 1-12 (2013). DOI: 10.1214/ECP.v18-2479


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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Daniel Valesin. Thomas Mountford. "Supercriticality of an annealed approximation of Boolean networks." Electron. Commun. Probab. 18 1 - 12, 2013.


Accepted: 4 May 2013; Published: 2013
First available in Project Euclid: 7 June 2016

zbMATH: 1302.82076
MathSciNet: MR3064991
Digital Object Identifier: 10.1214/ECP.v18-2479



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