Éminence Grise Coalitions: On the Shaping of Public Opinion

We consider an opinion network of multiple individuals with dynamics evolving via a general time-varying continuous time consensus algorithm. In such a network, a subset of individuals forms an éminence grise coalition (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 coupling graph of the network as well as the initial opinions of all other individuals. We establish the existence of a minimum size EGC and develop a nontrivial set of tight upper and lower bounds on that size. Thus, even when the coupling graph does not guarantee convergence to a global or multiple consensus, a generally restricted coalition of individuals can steer public opinion toward a desired consensus, provided they can cooperatively adjust their own initial opinions. Geometric insights into the structure of EGCs are also given.