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Supercriticality of an annealed approximation of Boolean networks

   
@article{ECP2479,
	author = {Daniel Valesin and Thomas Mountford},
	title = {Supercriticality of an annealed approximation of Boolean networks},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {18},
	year = {2013},
	keywords = {threshold contact process; random graphs; boolean networks},
	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 $n$.  
},
	pages = {no. 32, 1-12},
	issn = {1083-589X},
	doi = {10.1214/ECP.v18-2479},    
        url = {http://ecp.ejpecp.org/article/view/2479}}