Stability of the Kauffman Model

Random Boolean networks, the Kauffman model, are revisited by means of a novel decimation algorithm, which reduces the networks to their dynamical cores. The average size of the removed part, the stable core, grows approximately linearly with N, the number of nodes in the original networks. We show that this can be understood as the percolation of the stability signal in the network. The stability of the dynamical core is investigated and it is shown that this core lacks the well known stability observed in full Kauffman networks. We conclude that, somewhat counter-intuitive, the remarkable stability of Kauffman networks is generated by the dynamics of the 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 for that the number of limit cycles grows linearly with N. This result is in sharp contrast to the often cited $\sqrt{N}$ behavior.