Random Boolean Networks

Persistence of Activity in Random Boolean Networks

Shirshendu Chatterjee and Rick Durrett

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 site. Following a standard practice in the physics literature, we use a threshold contact process on a random graph 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 corresponds to chaotic behavior of the Boolean network. Unfortunately, we are only able to prove the persistence time is exp(cn^b(p)) with b(p) > 0 when 2p(1-p)r > 1 and b(p)=1 when 2p(1-p)(r-1) > 1.

Preprint

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