The effect of clusterings on the equilibrium states of local majority-rule: Occurrence probability and robustness

The equilibrium states associated with the local majority-rule are divided into three classes, the states of system-wide coordination, the trapped states, and the states of period-2. The effect of clustering coefficient on the occurrence probability of the states of three classes is analyzed numerically for Watts-Strogatz and scale-free networks. We further study the effect of clustering coefficient on the robustness for the states of each class by proposing a stochastic local majority-rule. The states of period-2 are found to be easy to break up, and the trapped states are most robust among the three classes. For systems in noisy environments, the proposed stochastic local majority-rule shows that there exists a range of noise for which, the mean first-passage time from strongly disorder states to the states of system-wide coordination is the shortest.