Supercriticality of an annealed approximation of Boolean networks

We consider a model recently proposed by Chatterjee and Durrett [1] as an "annealed approximation" of boolean networks, which are a class of cellular automata on a random graph, as defined by S. Kauffman [5]. The starting point is a random directed graph on n vertices; each vertex has r input vertices pointing to it. For the model of [1], 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 one of the results of [1], we show that if qr > 1, then the time of persistence of activity of the dynamics is exponential in n