  Consider a graph G with n nodes and m edges, which represents a social
network, and assume that initially each node is blue or white. In each round,
all nodes simultaneously update their color to the most frequent color in their
neighborhood. This is called the Majority Model (MM) if a node keeps its color
in case of a tie and the Random Majority Model (RMM) if it chooses blue with
probability 1/2 and white otherwise.
  We prove that there are graphs for which RMM needs exponentially many rounds
to reach a stable configuration in expectation, and such a configuration can
have exponentially many states (i.e., colorings). This is in contrast to MM,
which is known to always reach a stable configuration with one or two states in
$O(m)$ rounds. For the special case of a cycle graph C_n, we prove the stronger
and tight bounds of $\lceil n/2\rceil-1$ and $O(n^2)$ in MM and RMM,
respectively. Furthermore, we show that the number of stable colorings in MM on
C_n is equal to $\Theta(\Phi^n)$, where $\Phi = (1+\sqrt{5})/2$ is the golden
ratio, while it is equal to 2 for RMM.
  We also study the minimum size of a winning set, which is a set of nodes
whose agreement on a color in the initial coloring enforces the process to end
in a coloring where all nodes share that color. We present tight bounds on the
minimum size of a winning set for both MM and RMM.
  Furthermore, we analyze our models for a random initial coloring, where each
node is colored blue independently with some probability $p$ and white
otherwise. Using some martingale analysis and counting arguments, we prove that
the expected final number of blue nodes is respectively equal to
$(2p^2-p^3)n/(1-p+p^2)$ and pn in MM and RMM on a cycle graph C_n.
  Finally, we conduct some experiments which complement our theoretical
findings and also lead to the proposal of some intriguing open problems and
conjectures to be tackled in future work.
