Eminence Grise Coalitions: On the Shaping of Public Opinion

TL;DR

This paper investigates how a small coalition within a network of opinion dynamics can steer the entire network towards any desired consensus, providing bounds on the coalition size and geometric insights.

Contribution

It introduces the concept of eminence grise coalitions in opinion networks, establishing their existence, bounds on their size, and geometric properties, extending to discrete time cases.

Findings

A coalition of all individuals always forms an EGC.

Existence of a minimum size EGC is proven.

Bounds on the size of EGC are developed.

Abstract

We consider a network of evolving opinions. It includes multiple individuals with first-order opinion dynamics defined in continuous time and evolving based on a general exogenously defined time-varying underlying graph. In such a network, for an arbitrary fixed initial time, a subset of individuals forms an eminence grise coalition, abbreviated as EGC, if the individuals in that subset are capable of leading the entire network to agreeing on any desired opinion, through a cooperative choice of their own initial opinions. In this endeavor, the coalition members are assumed to have access to full profile of the underlying graph of the network as well as the initial opinions of all other individuals. While the complete coalition of individuals always qualifies as an EGC, we establish the existence of a minimum size EGC for an arbitrary time-varying network; also, we develop a non-trivial set of upper and lower bounds on that size. As a result, we show that, even when the underlying graph does not guarantee convergence to a global or multiple consensus, a generally restricted coalition of agents can steer public opinion towards a desired global consensus without affecting any of the predefined graph interactions, provided they can cooperatively adjust their own initial opinions. Geometric insights into the structure of EGC's are given. The results are also extended to the discrete time case where the relation with Decomposition-Separation Theorem is also made explicit.