Excitable Greenberg-Hastings cellular automaton model on scale-free networks

TL;DR

This paper analyzes the excitable Greenberg-Hastings cellular automaton on scale-free networks, deriving analytical expressions and exploring how activity and dynamic range depend on network topology and stimuli.

Contribution

It provides analytical solutions for the automaton's behavior on scale-free networks and links node connectivity to dynamic range, revealing insights into network topology effects.

Findings

The activity curve fits a Hill function with deviations.

The low-stimulus response follows a power law with exponent varying from 1 to 0.5.

Nodes with higher connectivity have larger optimal dynamic range.

Abstract

We study the excitable Greenberg-Hastings cellular automaton model on scale-free networks. We obtained analytical expressions for no external stimulus and the uncoupled case. It is found that the curves, the average activity $F$ versus the external stimulus rate $r$, can be fitted by a Hill function, but not exactly, and there exists a relation $F\propto r^\alpha$ for the low-stimulus response, where Stevens-Hill exponent $\alpha$ ranges from $\alpha = 1$ in the subcritical regime to $\alpha = 0.5$ at criticality. At the critical point, the range reaches the maximal. We also calculate the average activity $F^{k}(r)$ and the dynamic range $\Delta^{k}(p)$ for nodes with given connectivity $k$. It is interesting that nodes with larger connectivity have larger optimal range, which could be applied in biological experiments to reveal the network topology.