Activity patterns on random scale-free networks: global dynamics arising from localmajority rules

Abstract

Activity or spin patterns on a random scale-free network are studied using meanfield analysis and computer simulations. These activity patterns evolve in timeaccording to local majority rule dynamics which is implemented using (i)parallel or synchronous updating and (ii) random sequential or asynchronousupdating. Our mean field calculations predict that the relaxation processes ofdisordered activity patterns become much more efficient as the scaling exponentγ of the scale-free degree distribution changes fromγ > 5/2 toγ < 5/2.For γ > 5/2, the corresponding decay times increase asln(N) with increasingnetwork size N whereasthey are independent of N for γ < 5/2. In order to check these mean field predictions, extensive simulations of the patterndynamics have been performed using two different ensembles of random scale-freenetworks: (A) multi-networks as generated by the configuration method, whichtypically leads to many self-connections and multiple edges, and (B) simplenetworks without self-connections and multiple edges. We find that the mean fieldpredictions are confirmed (i) for random sequential updating of multi-networks and(ii) for both parallel and random sequential updating of simple networks withγ = 2.25 and2.6. For γ = 2.4, the data for the simple networks seem to be consistent with mean field theory as well,whereas we cannot draw a definite conclusion from the simulation data for the multi-networks.The latter difficulty can be understood in terms of an effective scaling exponentγeff = γeff(γ, N) for multi-networks. This effective exponent is determined byremoving all self-connections and multiple edges; it satisfiesγeff ≥ γ and decreases towardsγ with increasingnetwork size N. For γ = 2.4, we find γeff ≳ 5/2 up to N = 217.