## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 18 (2013), paper no. 32, 12 pp.

### Supercriticality of an annealed approximation of Boolean networks

Daniel Valesin and Thomas Mountford

#### Abstract

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 $n$

#### Article information

**Source**

Electron. Commun. Probab., Volume 18 (2013), paper no. 32, 12 pp.

**Dates**

Accepted: 4 May 2013

First available in Project Euclid: 7 June 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1465315571

**Digital Object Identifier**

doi:10.1214/ECP.v18-2479

**Mathematical Reviews number (MathSciNet)**

MR3064991

**Zentralblatt MATH identifier**

1302.82076

**Subjects**

Primary: Interacting Particle Systems

**Keywords**

threshold contact process random graphs boolean networks

**Rights**

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

#### Citation

Valesin, Daniel; Mountford, Thomas. Supercriticality of an annealed approximation of Boolean networks. Electron. Commun. Probab. 18 (2013), paper no. 32, 12 pp. doi:10.1214/ECP.v18-2479. https://projecteuclid.org/euclid.ecp/1465315571