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Jason Cawley
Wolfram Science Group
Phoenix, AZ USA

Registered: Aug 2003
Posts: 712

CA on network with asynchronous updates paper

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

Mihaela T. Matache and Jack Heidel

Physical Review E, published 28 February 2005

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 sARBNd in which only one node is updated at every time step, and for the class of generalized ARBNs sGARBNd 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.

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Old Post 06-10-2005 10:22 PM
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Jason Cawley
Wolfram Science Group
Phoenix, AZ USA

Registered: Aug 2003
Posts: 712

I thought this recent paper might be of interest to some here. It gets rather involved on the data analysis side, and is a bit overly restrictive in the rule examined. It can be hard to see the origin of interesting behavior in the case of randomized updating schemes, as well. But it is a worthwhile subject. Seth Chandler has shown CAs on regular networks (CANs) before, and others have done asynchronous rules, which can be viewed as a special case of mixed rules on one lattice (or network), with one of the rules in the mix being an identity rule.

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Old Post 06-10-2005 10:26 PM
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