  The "majority dynamics" process on a social network begins with an initial
phase, where the individuals are split into two competing parties, Red and
Blue. Every day, everyone updates their affiliation to match the majority among
those of their friends. While studying this process on Erdos-Renyi G(n, p)
random graph (with constant density), the authors discovered the "Power of Few"
phenomenon, showing that a very small advantage to one side already guarantees
that everybody will unanimously join that side after just a few days with
overwhelming probability. For example, when p = 1/2, then 10 extra members
guarantee this unanimity with a 90% chance, regardless of the value of n. In
this paper, we study this phenomenon for sparse random graphs. It is clear that
below the connectivity threshold, the phenomenon ceases to hold, as the
isolated vertices never change their colors. We show that it holds for every
density above the threshold. To make the process more realistic, we also assume
that individuals can randomly activate their accounts to post their opinions
and observe their neighbors (just as we do on social media). We prove that the
phenomenon is robust under this assumption.
