### Abstract

This study considers a simple Boolean network with N nodes, each node's state at time t being determined by a certain number of parent nodes. The network is analyzed when the connectivity k is fixed or variable. Making use of a Boolean rule that is a generalization of Rule 22 of elementary cellular automata, a generalized formula for providing the probability of finding a node in state 1 at time t is determined. We show typical behaviors of the iterations, and we study the dynamics of the network through Lyapunov exponents, bifurcation diagrams and fixed point analysis. We conclude that the network may exhibit stability or chaos, depending on the underlying parameters. In general high connectivity is associated with a convergence to zero of the probability of finding a node in state 1 at time t. We also study analytically and numerically the dynamics of the network under a stochastic noise procedure. We show that under a smaller probability of disturbing the nodes through the noise procedure the system tends to exhibit more nodes in the same state. For many parameter combinations there is no critical value of the noise parameter below which the network remains organized and above which it behaves randomly.

Original language | English (US) |
---|---|

Pages (from-to) | 4947-4958 |

Number of pages | 12 |

Journal | Physica A: Statistical Mechanics and its Applications |

Volume | 387 |

Issue number | 19-20 |

DOIs | |

State | Published - Aug 1 2008 |

### Fingerprint

### Keywords

- Dynamical phase transition
- Generalized elementary cellular automata rule 22
- Random Boolean network
- Stochastic noise
- System dynamics
- chaos

### ASJC Scopus subject areas

- Statistics and Probability
- Condensed Matter Physics

### Cite this

**Dynamical behavior and influence of stochastic noise on certain generalized Boolean networks.** / Beck, Gary L.; Matache, Mihaela T.

Research output: Contribution to journal › Article

*Physica A: Statistical Mechanics and its Applications*, vol. 387, no. 19-20, pp. 4947-4958. https://doi.org/10.1016/j.physa.2008.04.032

}

TY - JOUR

T1 - Dynamical behavior and influence of stochastic noise on certain generalized Boolean networks

AU - Beck, Gary L.

AU - Matache, Mihaela T

PY - 2008/8/1

Y1 - 2008/8/1

N2 - This study considers a simple Boolean network with N nodes, each node's state at time t being determined by a certain number of parent nodes. The network is analyzed when the connectivity k is fixed or variable. Making use of a Boolean rule that is a generalization of Rule 22 of elementary cellular automata, a generalized formula for providing the probability of finding a node in state 1 at time t is determined. We show typical behaviors of the iterations, and we study the dynamics of the network through Lyapunov exponents, bifurcation diagrams and fixed point analysis. We conclude that the network may exhibit stability or chaos, depending on the underlying parameters. In general high connectivity is associated with a convergence to zero of the probability of finding a node in state 1 at time t. We also study analytically and numerically the dynamics of the network under a stochastic noise procedure. We show that under a smaller probability of disturbing the nodes through the noise procedure the system tends to exhibit more nodes in the same state. For many parameter combinations there is no critical value of the noise parameter below which the network remains organized and above which it behaves randomly.

AB - This study considers a simple Boolean network with N nodes, each node's state at time t being determined by a certain number of parent nodes. The network is analyzed when the connectivity k is fixed or variable. Making use of a Boolean rule that is a generalization of Rule 22 of elementary cellular automata, a generalized formula for providing the probability of finding a node in state 1 at time t is determined. We show typical behaviors of the iterations, and we study the dynamics of the network through Lyapunov exponents, bifurcation diagrams and fixed point analysis. We conclude that the network may exhibit stability or chaos, depending on the underlying parameters. In general high connectivity is associated with a convergence to zero of the probability of finding a node in state 1 at time t. We also study analytically and numerically the dynamics of the network under a stochastic noise procedure. We show that under a smaller probability of disturbing the nodes through the noise procedure the system tends to exhibit more nodes in the same state. For many parameter combinations there is no critical value of the noise parameter below which the network remains organized and above which it behaves randomly.

KW - Dynamical phase transition

KW - Generalized elementary cellular automata rule 22

KW - Random Boolean network

KW - Stochastic noise

KW - System dynamics

KW - chaos

UR - http://www.scopus.com/inward/record.url?scp=44649088226&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=44649088226&partnerID=8YFLogxK

U2 - 10.1016/j.physa.2008.04.032

DO - 10.1016/j.physa.2008.04.032

M3 - Article

VL - 387

SP - 4947

EP - 4958

JO - Physica A: Statistical Mechanics and its Applications

JF - Physica A: Statistical Mechanics and its Applications

SN - 0378-4371

IS - 19-20

ER -