Discrete Opinion Dynamics with Stubborn Agents

We study discrete opinion dynamics in a social network with "stubborn agents" who influence others but do not change their opinions. We generalize the classical voter model by introducing nodes (stubborn agents) that have a fixed state. We show that the presence of stubborn agents with opposing opinions precludes convergence to consensus; instead, opinions converge in distribution with disagreement and fluctuations. In addition to the first moment of this distribution typically studied in the literature, we study the behavior of the second moment in terms of network properties and the opinions and locations of stubborn agents. We also study the problem of "optimal placement of stubborn agents" where the location of a fixed number of stubborn agents is chosen to have the maximum impact on the long-run expected opinions of agents