Activity patterns on random scale-free networks: Global dynamics arising from local majority rules

TL;DR

This study investigates how activity patterns on scale-free networks evolve under local majority rules, revealing that the relaxation efficiency depends on the network's degree distribution exponent, with different behaviors for g > 5/2 and g < 5/2.

Contribution

The paper combines mean field analysis and simulations to show how the relaxation dynamics on scale-free networks depend on the degree distribution exponent g, providing new insights into global behavior from local rules.

Findings

Relaxation times grow as ln(N) for g > 5/2.

Relaxation times are independent of N for g < 5/2.

Mean field predictions are validated by extensive simulations.

Abstract

Activity or spin patterns on random scale-free network are studied by mean field analysis and computer simulations. These activity patterns evolve in time according to local majority-rule dynamics which is implemented using (i) parallel or synchronous updating and (ii) random sequential or asynchronous updating. Our mean-field calculations predict that the relaxation processes of disordered activity patterns become much more efficient as the scaling exponent $\gamma$ of the scale-free degree distribution changes from $\gamma >5/2$ to $\gamma < 5/2$. For $\gamma > 5/2$, the corresponding decay times increase as $\ln(N)$ with increasing network size $N$ whereas they are independent of $N$ for $\gamma < 5/2$. In order to check these mean field predictions, extensive simulations of the pattern dynamics have been performed using two different ensembles of random scale-free networks: (A) multi-networks as generated by the configuration method, which typically leads to many self-connections and multiple edges, and (B) simple-networks without self-connections and multiple edges.