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

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

2009-11-30

arxiv.org/abs/0911.5339v1

Mathematics

Probability

21 pages

Scientific paper

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 $r\ge 3$ 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 $(r-1)\cdot 2p(1-p)>1$.

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