Opinion fluctuations and disagreement in social networks

We study a tractable opinion dynamics model that generates long-run disagreements and persistent opinion fluctuations. Our model involves a inhomogeneous stochastic gossip process of continuous opinion dynamics in a society consisting of two types of agents: regular agents, who update their beliefs according to information that they receive from their social neighbors; and stubborn agents, who never update their opinions and might represent leaders, political parties or media sources attempting to influence the beliefs in the rest of the society. When the society contains stubborn agents with different opinions, the belief dynamics never lead to a consensus (among the regular agents). Instead, beliefs in the society almost surely fail to converge, the belief profile keeps on oscillating in an ergodic fashion, and it converges in law to a non-degenerate random vector. The structure of the graph describing the social network and the location of stubborn agents within it shape the opinion dynamics. The expected belief vector is proved to evolve according to an ordinary differential equation coinciding with the Kolmogorov backward equation of a continuos time Markov chain on the graph with absorbing states corresponding to the stubborn agents, and hence to converge to a harmonic vector, with every regular agent’s value being the weighted average of its neighbors ’ values, and boundary conditions corresponding to the stubborn agents ’ beliefs. Expected cross-products of the stationary beliefs allow for a similar characterization in terms of coupled random walks on the graph describing the social network