Opinion dynamics and the evolution of social power in influence networks.Jia, P., MirTabatabaei, A., Friedkin, N.E. and 1 more (2015) SIAM Review, 57 (3), pp. 367-397.

This paper studies the evolution of self-appraisal, social power, and interpersonal influences for a group of individuals who discuss and form opinions about a sequence of issues. Our empirical model combines the averaging rule of DeGroot to describe opinion formation processes and the reflected appraisal mechanism of Friedkin to describe the dynamics of individuals' self-appraisal and social power. Given a set of relative interpersonal weights, the DeGroot--Friedkin model predicts the evolution of the influence network governing the opinion formation process. We provide a rigorous mathematical formulation of the influence network dynamics, characterize its equilibria, and establish its convergence properties for all possible structures of the relative interpersonal weights and corresponding eigenvector centrality scores. The model predicts that the social power ranking among individuals is asymptotically equal to their centrality ranking, that social power tends to accumulate at the top of the hierarchy, and that an autocratic (resp., democratic) power structure arises when the centrality scores are maximally nonuniform (resp., uniform).