Convergence, unanimity and disagreement in majority dynamics on unimodular graphs and random graphs

In majority dynamics, agents located at the vertices of an undirected simple graph update their binary opinions synchronously by adopting those of the majority of their neighbors. On infinite unimodular transitive graphs we show that the opinion of each agent almost surely either converges, or else eventually oscillates with period two; this is known to hold for finite graphs, but not for all infinite graphs. On Erdős–Rényi random graphs with degrees Ω(n), we show that agents eventually all agree, with constant probability. Conversely, on random 4-regular finite graphs, we show that with high probability different agents converge to different opinions.