Stability of the Kauffman model

Random Boolean networks, the Kauffman model, are revisited by means of a Nobel decimation algorithm, which removes variables that cannot be relevant to the asymptotic dynamics of the system. The major part of the removed variables have the same fixed state in all limit cycles. These variables are denoted as the stable core of the network and their number grows approximately linearly with N, the number of variables in the original network. The sensitivity of the attractors to perturbations is investigated. We find that reduced networks lack the well-known insensitivity observed in full Kauffman networks. We conclude that, somewhat counterintuitive, this remarkable property of full Kauffman networks is generated by the dynamics of their stable core. The decimation method is also used to simulate large critical Kauffman networks. For networks up to N = 32 we perform full enumeration Studies. Strong evidence is provided that the number of limit cycles grows linearly with N. This result is in sharp contrast to the often cited behavior.