Mean-field and non-mean-field behaviors in scale-free networks with random boolean dynamics.

Abstract

We study two types of simplified Boolean dynamics in scale-free networks, both with synchronous update. Assigning only Boolean functions AND and XOR to the nodes with probability 1 − p and p, respectively, we are able to analyze the density of 1’s and the Hamming distance on the network by numerical simulations and by a mean-field approximation (annealed approximation). We show that the behavior is quite different if the node always enters in the dynamics as its own input (self-regulation) or not. The same conclusion holds for the Kauffman NK model. Moreover, the simulation results and the mean-field ones (i) agree well when there is no self-regulation, and (ii) disagree for small p when self-regulation is present in the model.