The "Power of Few" Phenomenon: The Sparse Case

TL;DR

This paper investigates the 'Power of Few' phenomenon in social networks, demonstrating that a small initial advantage can lead to complete consensus, even in sparse graphs, and shows this effect is robust under realistic social media dynamics.

Contribution

It extends the understanding of the 'Power of Few' phenomenon to sparse random graphs and incorporates realistic account activation, showing the effect persists under these conditions.

Findings

Small initial advantage guarantees consensus above the connectivity threshold.

The phenomenon does not hold below the connectivity threshold due to isolated vertices.

Robustness of the phenomenon under random account activation.

Abstract

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.