### Persistence of Activity in Threshold Contact Processes, an "Annealed
Approximation" of Random Boolean Networks

Shirshendu Chatterjee

Abstract.
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 is at least 3 and 2p(1-p)r>1, then the
threshold contact process persists for a long time, which correspond
to chaotic behavior of the Boolean network. Joint work with Rick
Durrett.