# 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..

### Full Text

### Duke Authors

### Cited Authors

- Chatterjee, S; Durrett, R

### Published Date

- September 1, 2011

### Published In

### Volume / Issue

- 39 / 2

### Start / End Page

- 228 - 246

### Electronic International Standard Serial Number (EISSN)

- 1098-2418

### International Standard Serial Number (ISSN)

- 1042-9832

### Digital Object Identifier (DOI)

- 10.1002/rsa.20357

### Citation Source

- Scopus