### Abstract

This paper considers a simple Boolean network with N nodes, each node's state at time t being determined by a certain number k of parent nodes, which is fixed for all nodes. The nodes, with randomly assigned neighborhoods, are updated based on various asynchronous schemes. We make use of a Boolean rule that is a generalization of rule 126 of elementary cellular automata. We provide formulas for the probability of finding a node in state 1 at a time t for the class of asynchronous random Boolean networks (ARBN) in which only one node is updated at every time step, and for the class of generalized ARBNs (GARBN) in which a random number of nodes can be updated at each time point. We use simulation methods to generate consecutive states of the network for both the real system and the models under the various schemes. The results match well. We study the dynamics of the models through sensitivity of the orbits to initial values, bifurcation diagrams, and fixed point analysis. We show, both theoretically and by example, that the ARBNs generate an ordered behavior regardless of the updating scheme used, whereas the GARBNs have behaviors that range from order to chaos depending on the type of random variable used to determine the number of nodes to be updated and the parameter combinations.

Original language | English (US) |
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Article number | 026232 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 71 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1 2005 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability
- Condensed Matter Physics

### Cite this

**Asynchronous random Boolean network model based on elementary cellular automata rule 126.** / Matache, Mihaela T.; Heidel, Jack.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Asynchronous random Boolean network model based on elementary cellular automata rule 126

AU - Matache, Mihaela T.

AU - Heidel, Jack

PY - 2005/2/1

Y1 - 2005/2/1

N2 - This paper considers a simple Boolean network with N nodes, each node's state at time t being determined by a certain number k of parent nodes, which is fixed for all nodes. The nodes, with randomly assigned neighborhoods, are updated based on various asynchronous schemes. We make use of a Boolean rule that is a generalization of rule 126 of elementary cellular automata. We provide formulas for the probability of finding a node in state 1 at a time t for the class of asynchronous random Boolean networks (ARBN) in which only one node is updated at every time step, and for the class of generalized ARBNs (GARBN) in which a random number of nodes can be updated at each time point. We use simulation methods to generate consecutive states of the network for both the real system and the models under the various schemes. The results match well. We study the dynamics of the models through sensitivity of the orbits to initial values, bifurcation diagrams, and fixed point analysis. We show, both theoretically and by example, that the ARBNs generate an ordered behavior regardless of the updating scheme used, whereas the GARBNs have behaviors that range from order to chaos depending on the type of random variable used to determine the number of nodes to be updated and the parameter combinations.

AB - This paper considers a simple Boolean network with N nodes, each node's state at time t being determined by a certain number k of parent nodes, which is fixed for all nodes. The nodes, with randomly assigned neighborhoods, are updated based on various asynchronous schemes. We make use of a Boolean rule that is a generalization of rule 126 of elementary cellular automata. We provide formulas for the probability of finding a node in state 1 at a time t for the class of asynchronous random Boolean networks (ARBN) in which only one node is updated at every time step, and for the class of generalized ARBNs (GARBN) in which a random number of nodes can be updated at each time point. We use simulation methods to generate consecutive states of the network for both the real system and the models under the various schemes. The results match well. We study the dynamics of the models through sensitivity of the orbits to initial values, bifurcation diagrams, and fixed point analysis. We show, both theoretically and by example, that the ARBNs generate an ordered behavior regardless of the updating scheme used, whereas the GARBNs have behaviors that range from order to chaos depending on the type of random variable used to determine the number of nodes to be updated and the parameter combinations.

UR - http://www.scopus.com/inward/record.url?scp=41349120085&partnerID=8YFLogxK

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U2 - 10.1103/PhysRevE.71.026232

DO - 10.1103/PhysRevE.71.026232

M3 - Article

C2 - 15783412

AN - SCOPUS:41349120085

VL - 71

JO - Physical review. E

JF - Physical review. E

SN - 1539-3755

IS - 2

M1 - 026232

ER -