Persistence of activity in threshold contact processes, an “Annealed approximation” of random Boolean networks

We consider a model for gene regulatory networks that is a modification of Kauffmann's (1969) random Boolean networks. There are three parameters: $n =$ the number of nodes, $r =$ the number of inputs to each node, and $p =$ the expected fraction of 1's in the Boolean functions at each node. Following a standard practice in the physics literature, we use a threshold contact process on a random graph on $n$ nodes, in which each node has in degree $r$, to approximate its dynamics. We show that if $r\ge 3$ and $r \cdot 2p(1-p)>1$, then the threshold contact process persists for a long time, which correspond to chaotic behavior of the Boolean network. Unfortunately, we are only able to prove the persistence time is $\ge \exp(cn^{b(p)})$ with $b(p)>0$ when $r\cdot 2p(1-p)> 1$, and $b(p)=1$ when $(r-1)\cdot 2p(1-p)>1$.