Supercriticality of an annealed approximation of Boolean networks

Daniel Valesin (University of British Columbia)
Thomas Mountford (École Polytechnique Fédérale de Lausanne)


We consider a model recently proposed by Chatterjee and Durrett as an "annealed approximation'' of boolean networks, which are a class of cellular automata on a random graph, as defined by S. Kauffman. The starting point is a random directed graph on $n$ vertices; each vertex has $r$ input vertices pointing to it. For the model of Chatterjee and Durrett, a discrete time threshold contact process is then considered on this graph: at each instant, each vertex has probability $q$ of choosing to receive input; if it does, and if at least one of its input vertices were in state 1 at the previous instant, then it is labelled with a 1; in all other cases, it is labelled with a 0. $r$ and $q$ are kept fixed and $n$ is taken to infinity. Improving a result of Chatterjee and Durrett, we show that if $qr > 1$, then the time of persistence of activity of the dynamics is exponential in $n$.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-12

Publication Date: May 4, 2013

DOI: 10.1214/ECP.v18-2479


  • Chatterjee, Shirshendu; Durrett, Rick. Persistence of activity in threshold contact processes, an "annealed approximation'' of random Boolean networks. Random Structures Algorithms 39 (2011), no. 2, 228--246. MR2850270
  • B. Derrida, Y. Pomeau, Random networks of automata: a simple annealed approximation, Europhysics Letters 1 (1986), 45-49
  • Durrett, Rick. Probability: theory and examples. Fourth edition. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010. x+428 pp. ISBN: 978-0-521-76539-8 MR2722836
  • Durrett, Rick. Random graph dynamics. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2007. x+212 pp. ISBN: 978-0-521-86656-9; 0-521-86656-1 MR2271734
  • S. Kauffman, Metabolic stability and epigenesis in randomly constructed genetic nets. Journal of Theoretical Biology 22 (1969), 437-467
  • S. A. Kauffman, Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press, 1993.

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.