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

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..

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## Related Subject Headings

- Computation Theory & Mathematics
- 4904 Pure mathematics
- 4901 Applied mathematics
- 4613 Theory of computation
- 0802 Computation Theory and Mathematics
- 0104 Statistics
- 0101 Pure Mathematics

### Citation

*Random Structures and Algorithms*,

*39*(2), 228–246. https://doi.org/10.1002/rsa.20357

*Random Structures and Algorithms*39, no. 2 (September 1, 2011): 228–46. https://doi.org/10.1002/rsa.20357.

*Random Structures and Algorithms*, vol. 39, no. 2, Sept. 2011, pp. 228–46.

*Scopus*, doi:10.1002/rsa.20357.

## Published In

## DOI

## EISSN

## ISSN

## Publication Date

## Volume

## Issue

## Start / End Page

## Related Subject Headings

- Computation Theory & Mathematics
- 4904 Pure mathematics
- 4901 Applied mathematics
- 4613 Theory of computation
- 0802 Computation Theory and Mathematics
- 0104 Statistics
- 0101 Pure Mathematics