Mean-Field and Non-Mean-Field Behaviors in Scale-free Networks with Random Boolean Dynamics

Abstract

We study two types of simplified Boolean dynamics over 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 dynamic as its own input (self-regulation) or not. The same conclusion holds for the Kauffman KN 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.

12 pages, 7 figures

References in corpus (4)

• Evolution of networks
• Kauffman networks with threshold functions
• Critical Boolean networks with scale-free in-degree distribution
• Broad edge of chaos in strongly heterogeneous Boolean networks