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The equilibrium states associated with the local majority-rule are dividedinto three classes, the states of system-wide coordination, the trapped states,and the states of period-2. The effect of clustering coefficient on theoccurrence probability of the states of three classes is analyzed numericallyfor Watts-Strogatz and scale-free networks. We further study the effect ofclustering coefficient on the robustness for the states of each class byproposing a stochastic local majority-rule. The states of period-2 are found tobe easy to break up, and the trapped states are most robust among the threeclasses. For systems in noisy environments, the proposed stochastic localmajority-rule shows that there exists a range of noise for which, the meanfirst-passage time from strongly disorder states to the states of system-widecoordination is the shortest.

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