  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.
