Random maps and attractors in random Boolean networks

Abstract

Despite their apparent simplicity, random Boolean networks display a rich variety of dynamical behaviors. Much work has been focused on the properties and abundance of attractors. The topologies of random Boolean networks with one input per node can be seen as graphs of random maps. We introduce an approach to investigating random maps and finding analytical results for attractors in random Boolean networks with the corresponding topology. Approximating some other non-chaotic networks to be of this class, we apply the analytic results to them. For this approximation, we observe a strikingly good agreement on the numbers of attractors of various lengths. We also investigate observables related to the average number of attractors in relation to the typical number of attractors. Here, we find strong differences that highlight the difficulties in making direct comparisons between random Boolean networks and real systems. Furthermore, we demonstrate the power of our approach by deriving some results for random maps. These results include the distribution of the number of components in random maps, along with asymptotic expansions for cumulants up to the 4th order.

18 pages, 11 figures

References in corpus (6)

• Random Boolean Network Models and the Yeast Transcriptional Network
• Genetic networks with canalyzing Boolean rules are always stable
• Number and length of attractors in a critical Kauffman model with connectivity one
• Scaling in ordered and critical random Boolean networks
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Cited by in corpus (3)

• Canalization and Symmetry in Boolean Models for Genetic Regulatory Networks
• Relevant components in critical random Boolean networks
• Periodic attractors of random truncator maps