Minority Becomes Majority in Social Networks

Abstract

It is often observed that agents tend to imitate the behavior of their neighbors in a social network. This imitating behavior might lead to the strategic decision of adopting a public behavior that differs from what the agent believes is the right one and this can subvert the behavior of the population as a whole.

In this paper, we consider the case in which agents express preferences over two alternatives and model social pressure with the majority dynamics: at each step an agent is selected and its preference is replaced by the majority of the preferences of her neighbors. In case of a tie, the agent does not change her current preference. A profile of the agents’ preferences is stable if the each agent’s preference coincides with the preference of at least half of the neighbors (thus, the system is in equilibrium).

We ask whether there are network topologies that are robust to social pressure. That is, we ask whether there are graphs in which the majority of preferences in an initial profile ${\mathbf {s}}$ always coincides with the majority of the preference in all stable profiles reachable from ${\mathbf {s}}$. We completely characterize the graphs with this robustness property by showing that this is possible only if the graph has no edge or is a clique or very close to a clique. In other words, except for this handful of graphs, every graph admits at least one initial profile of preferences in which the majority dynamics can subvert the initial majority. We also show that deciding whether a graph admits a minority that becomes majority is NP-hard when the minority size is at most 1 / 4-th of the social network size.

This work was partially supported by the COST Action IC1205 “Computational Social Choice”, by the Italian MIUR under the PRIN 2010–2011 project ARS TechnoMedia – Algorithmics for Social Technological Networks, by the European Social Fund and Greek national funds through the research funding program Thales on “Algorithmic Game Theory”, by the EU FET project MULTIPLEX 317532, and by a Caratheodory basic research grant from the University of Patras.