Opinion forming in Erdős-Rényi random graph and expanders.

Assume for a graph G = (V, E) and an initial configuration, where each node is blue
or red, in each discrete-time round all nodes simultaneously update their color to the
most frequent color in their neighborhood and a node keeps its color in case of a tie.
We study the behavior of this basic process, which is called the majority model, on the
Erdős–Rényi random graph Gn,p and regular expanders. First we consider the behavior
of the majority model on Gn,p with an initial random configuration, where each node
is blue independently with probability pb and red otherwise. It is shown that in this
setting the process goes through a phase transition at the connectivity threshold, namely
log n/n. Furthermore, we say a graph G is λ-expander if the second-largest absolute
eigenvalue of its adjacency matrix is λ. We prove that for a ∆-regular λ-expander graph
if λ/∆ is sufficiently small, then the majority model by starting from (1/2 − δ) n blue
nodes (for an arbitrarily small constant δ > 0) results in fully red configuration in sublogarithmically many rounds. Roughly speaking, this means the majority model is an
‘‘efficient’’ and ‘‘fast’’ density classifier on regular expanders. As a by-product of our
results, we show that regular Ramanujan graphs are asymptotically optimally immune,
that is for an n-node ∆-regular Ramanujan graph if the initial number of blue nodes is
s ≤ βn, the number of blue nodes in the next round is at most cs/∆ for some constants
c, β > 0