Eminence grise coalitions in opinion dynamics

In this paper, we consider linear continuous-time models of evolution of opinions in large-scale dynamical networks. Our focus is on dynamical networks that are defined on general exogenously given time-varying graphs, where the nodes of the underlying graph model individuals (or agents) with first-order linear opinion dynamics. In such a network, for an arbitrary fixed initial time, a subset of individuals is said to form an eminence grise coalition (EGC) if its members are capable of leading the entire network to agreeing on any desired value through a cooperative choice of their own initial opinions. 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 in an opinion network trivially qualifies as an EGC, we establish the existence of a minimum size EGC for an arbitrary time-varying network and 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 single or multiple consensus, a generally restricted coalition of agents can steer public opinion towards a desired consensus without affecting any of the predefined graph interactions by having its members collectively adjust their own initial opinions. Geometric insights into the structure of EGC´s are also developed.