Dynamics of Majority Rule in Two-State Interacting Spin Systems

We introduce a two-state opinion dynamics model where agents evolve by majority rule. In each update, a group of agents is specified whose members then all adopt the local majority state. In the mean-field limit, where a group consists of randomly selected agents, consensus is reached in a time that scales $\ln N$, where $N$ is the number of agents. On finite-dimensional lattices, where a group is a contiguous cluster, the consensus time fluctuates strongly between realizations and grows as a dimension-dependent power of $N$. The upper critical dimension appears to be larger than 4. The final opinion always equals that of the initial majority except in one dimension.