Reducing Polarization in Opinion Networks in the Presence of Stubborn Leaders

Abstract: We study the problem of reducing polarization (variance) of opinions at stationarity in a directed weighted graph with node set divided into two groups: stubborn, initialized with a fixed opinion and regular who repeatedly update their opinion to the average of their out-neighbors, known as the DeGroot model with stubborn nodes. We show how the polarization can be minimized for a number of simple constraints, but that the problem in general is not convex. Theory is developed for the change in opinions at stationarity and the polarization measure for a rank-1 update of the network (encompassing both addition of a directed and undirected link in the network). An algorithm for gradient approximation is presented, given directly by the analytical gradient formulation and method of matrix-vector product estimation. Lastly variations of the algorithm together with other trivial methods of recommending a link are compared for a number of random and real networks.