  We consider a model for heterogeneous 'gene regulatory networks' that is a
generalization of the model proposed by Chatterjee and Durrett (2011) as an
"annealed approximation" of Kauffmann's (1969) random Boolean networks. In this
model, genes are represented by the nodes of a random directed graph on n
vertices with specified in-degree distribution (resp. out-degree distribution
or joint distribution of in-degree and out-degree), and the expression bias
(the expected fraction of 1's in the Boolean functions) p is same for all
nodes. Following a standard practice in the physics literature, we use a
discrete-time threshold contact process with parameter q=2p(1-p) (in which a
vertex with at least one 'occupied' input at time t will be occupied at time
t+1 with probability q, and 'vacant' otherwise) on the above random graph to
approximate the dynamics of the Boolean network. We show that there is a
parameter r (which can be written explicitly in terms of first few moments of
the degree distribution) such that, with probability tending to 1 as n goes to
infinity, if 2p(1-p)r>1, then starting from all occupied sites the threshold
contact process maintains a positive ({\it quasi-stationary}) density of
occupied sites for time which is exponential in n, whereas if 2p(1-p)r<1, then
the persistence time of the threshold contact process is at most logarithmic in
n. These two phases correspond to the 'chaotic' and 'ordered' behavior of the
gene networks.
