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 J Theor Biol 22 (1969), 437–467 random Boolean networks. There are three parameters: $n = {rm the}$ number of nodes, $r = {rm the}$ number of inputs to each node, and $p = {rm the}$ expected fraction of 1'sin the Boolean functions at each node. Following a standard practice in thephysics 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 $rge 3$ 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 $rcdot 2p(1-p)> 1$ , and $b(p)=1$ when $(r-1)cdot 2p(1-p)>1$ . © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011