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

TL;DR

This paper investigates Boolean dynamics on scale-free networks, comparing mean-field and non-mean-field behaviors, especially focusing on the effects of self-regulation and different Boolean functions, through simulations and analytical approximations.

Contribution

It introduces a comparative analysis of Boolean dynamics with and without self-regulation on scale-free networks, highlighting differences in behavior and approximation accuracy.

Findings

Mean-field and simulation results agree well without self-regulation.

Discrepancies occur at small p with self-regulation present.

Self-regulation significantly affects network 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.