Supercriticality of Annealed Approximations of Boolean Networks

We consider a model recently proposed by Chatterjee and Durrett [CD2011] as an "annealed approximation" of boolean networks, which are a class of cellular automata on a random graph, as defined by S. Kauffman [K69]. The starting point is a random directed graph on $n$ vertices; each vertex has $r$ input vertices pointing to it. For the model of [CD2011], 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 [CD2011], we show that if $qr > 1$, then the time of persistence of activity of the dynamics is exponential in $n$.