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

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.