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

Publication ,  Journal Article
Chatterjee, S; Durrett, R
Published in: Random Structures and Algorithms
September 1, 2011

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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Published In

Random Structures and Algorithms

DOI

EISSN

1098-2418

ISSN

1042-9832

Publication Date

September 1, 2011

Volume

39

Issue

2

Start / End Page

228 / 246

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

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Chatterjee, S., & Durrett, R. (2011). Persistence of activity in threshold contact processes, an "Annealed approximation" of random Boolean networks. Random Structures and Algorithms, 39(2), 228–246. https://doi.org/10.1002/rsa.20357
Chatterjee, S., and R. Durrett. “Persistence of activity in threshold contact processes, an "Annealed approximation" of random Boolean networks.” Random Structures and Algorithms 39, no. 2 (September 1, 2011): 228–46. https://doi.org/10.1002/rsa.20357.
Chatterjee S, Durrett R. Persistence of activity in threshold contact processes, an "Annealed approximation" of random Boolean networks. Random Structures and Algorithms. 2011 Sep 1;39(2):228–46.
Chatterjee, S., and R. Durrett. “Persistence of activity in threshold contact processes, an "Annealed approximation" of random Boolean networks.” Random Structures and Algorithms, vol. 39, no. 2, Sept. 2011, pp. 228–46. Scopus, doi:10.1002/rsa.20357.
Chatterjee S, Durrett R. Persistence of activity in threshold contact processes, an "Annealed approximation" of random Boolean networks. Random Structures and Algorithms. 2011 Sep 1;39(2):228–246.
Journal cover image

Published In

Random Structures and Algorithms

DOI

EISSN

1098-2418

ISSN

1042-9832

Publication Date

September 1, 2011

Volume

39

Issue

2

Start / End Page

228 / 246

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