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Dynamics of Random Boolean Networks with Constant Number of Parents and Asynchrony Generated by Stochastic Processes

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 asynchronously updated based on various stochastic processes. We make use of a Boolean rule that is a generalization of rule 126 of elementary cellular automata. We provide the formula for the probability of finding a node in state 1 at a time t for the class of Generalized Asynchronous Random Boolean Networks (GARBNs) in which a random number of nodes can be updated at each time point. Poisson and random walk processes are used as the random number generators. We use simulation methods to generate consecutive states of the network for both the real system and the models under the various parameter combinations and processes. 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 that the GARBNs have behaviors that range from order to chaos depending on the type and the parameter combinations of the stochastic processes used to determine the number of nodes to be updated. We observe that for some parameter combinations, the system becomes eventually synchronous. We explain this behavior and the associated study of the dynamics.