Number and length of attractors in a critical Kauffman model with connectivity one

Abstract

The Kauffman model describes a system of randomly connected nodes with dynamics based on Boolean update functions. Though it is a simple model, it exhibits very complex behavior for "critical" parameter values at the boundary between a frozen and a disordered phase, and is therefore used for studies of real network problems. We prove here that the mean number and mean length of attractors in critical random Boolean networks with connectivity one both increase faster than any power law with network size. We derive these results by generating the networks through a growth process and by calculating lower bounds.

4 pages, no figure, no table; published in PRL

References in corpus (5)

• Random Boolean Network Models and the Yeast Transcriptional Network
• Scaling in ordered and critical random Boolean networks
• Stability of the Kauffman Model
• Relevant elments, Magnetization and Dynamical Properties in Kauffman Networks: a Numerical Study
• The Modular Structure of Kauffman Networks

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