  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 (e.g., Cayley graphs), when initial
opinions are chosen from a distribution that is invariant with respect to the
graph automorphism group, 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 Erd\H{o}s-R\'enyi random graphs with degrees $\Omega(\sqrt{n})$, we show
that when initial opinions are chosen i.i.d. then agents all converge to the
initial majority opinion, with constant probability. Conversely, on random
4-regular finite graphs, we show that with high probability different agents
converge to different opinions.
