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

Published

Journal Article

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 $r\ge 3$ 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 $(r-1)\cdot 2p(1-p)>1$. © 2011 Wiley Periodicals, Inc..

Cited Authors

• Chatterjee, S; Durrett, R

Published Date

• September 1, 2011

• 39 / 2

• 228 - 246

• 1098-2418

• 1042-9832

Digital Object Identifier (DOI)

• 10.1002/rsa.20357

• Scopus