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

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.

💡 Research Summary

The paper investigates random Boolean dynamics on scale‑free networks using two simplified models that differ only in whether a node’s own previous state is included as an input (self‑regulation). In both models each node updates synchronously and applies either an AND or an XOR Boolean function to its inputs. The AND function is chosen with probability 1‑p and the XOR function with probability p, making p the control parameter that tunes the degree of non‑linearity and potential chaos in the system. The authors study two macroscopic observables: the density of nodes in state 1, M(t), and the Hamming distance D(t) between two initially close configurations.

A mean‑field (annealed) approximation is derived under the assumption that the inputs to each node are statistically independent at each time step. For the case without self‑regulation the recursion for the density reads
M(t+1) = (1‑p)