  This paper focuses on Majority Dynamics in sparse graphs, in particular, as a
tool to study internal cuts. It is known that, in Majority Dynamics on a finite
graph, each vertex eventually either comes to a fixed state, or oscillates with
period two. The empirical evidence acquired by simulations suggests that for
random odd-regular graphs, approximately half of the vertices end up
oscillating with high probability. We notice a local symmetry between
oscillating and non-oscillating vertices, that potentially can explain why the
fraction of the oscillating vertices is concentrated around $\frac{1}{2}$. In
our simulations, we observe that the parts of random odd-regular graph under
Majority Dynamics with high probability do not contain $\lceil \frac{d}{2}
\rceil$-cores at any timestep, and thus, one cannot use Majority Dynamics to
prove that internal cuts exist in odd-regular graphs almost surely. However, we
suggest a modification of Majority Dynamics, that yields parts with desired
cores with high probability.
