Consensus states of local majority rule in stochastic process

•A sufficient condition for reaching a consensus.•The relation between the geometry of networks and the reachability of a consensus.•Stochastic local majority rule.•The mean first-passage time and the escape rate of consensus states.

A sufficient condition for a network system to reach a consensus state of the local majority rule is shown. The influence of interpersonal environment on the occurrence probability of consensus states for Watts–Strogatz and scale-free networks with random initial states is analyzed by numerical method. We also propose a stochastic local majority rule to study the mean first passage time from a random state to a consensus and the escape rate from a consensus state for systems in a noisy environment. Our numerical results show that there exists a window of fluctuation strengths for which the mean first passage time from a random to a consensus state reduces greatly, and the escape rate of consensus states obeys the Arrhenius equation in the window.