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

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

2009-10-27

arxiv.org/abs/0910.5206v2

J. Phys. A: Math. Theor. 43, 225101 (18pp) (2010)

Physics

Biological Physics

12 pages, 7 figures

Scientific paper

10.1088/1751-8113/43/22/225101

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.

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