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

We study two types of simplified Boolean dynamics in scale-free networks, both with a synchronous update. Assigning only the Boolean functions AND and XOR to the nodes with probabilities 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.