Math @ Duke

Publications [#243523] of Richard T. Durrett
Papers Published
 Chatterjee, S; Durrett, R, Persistence of activity in threshold contact processes, an "Annealed approximation" of random Boolean networks,
Random Structures and Algorithms, vol. 39 no. 2
(2009),
pp. 228246, ISSN 10429832 [MR2850270], [doi]
(last updated on 2017/12/16)
Abstract: We consider a model for gene regulatory networks that is a modification of Kauffmann's J Theor Biol 22 (1969), 437467 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 $r \cdot 2p(1p)>1$, 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 $r\cdot 2p(1p)> 1$, and $b(p)=1$ when $(r1)\cdot 2p(1p)>1$. © 2011 Wiley Periodicals, Inc..


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