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 by DeGroot to describe opinion formation processes and the reflected appraisal mechanism by 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 non-uniform (resp. uniform).

This article reports new advancements in the theory of influence system evolution in small deliberative groups, and a novel set of empirical findings on such evolution. The theory elaborates the specification of the single-issue opinion dynamics of such groups, which has been the focus of theory development in the field of opinion dynamics, to include group dynamics that occur along a sequence of issues. The theory predicts an evolution of influence centralities along issue sequences based on elementary reflected appraisal mechanisms that modify influence network structure and flows of influence in the group. The new empirical findings, which are also reported in this article, present a remarkable suite of issue-sequence effects on influence network structure consistent with theoretical predictions.

Abstract-This article focuses on the evolution of interpersonal influences in a group of stubborn individuals as they discuss a sequence of issues. Each individual opinion about a single issue is updated based upon the convex combination of the individual's current opinion, the neighbors' current opinion, and the individual's initial opinion; the attachment to the initial opinion characterizes how stubborn an individual is. To model the evolution of the influence network, we employ Friedkin's "reflected appraisal" model: each individual's selfweight on a new issue is determined by the individual's average influence and relative control on other individuals on prior issue outcomes. These modeling assumptions lead to a dynamical system for the evolution of self-weights. We establish the well-posedness and continuity of the proposed dynamics and prove the existence and uniqueness of equilibria for stubborn individuals. We then study the impact of network topology on the individuals' final self-weights. We prove the convergence of all system trajectories for the special case of doubly-stochastic networks and homogeneous stubbornness. We characterize equilibrium self-weights for systems with centralized networks and heterogeneous stubbornness. Finally, our numerical simulations illustrate how existence, uniqueness and attractivity of the equilibria holds true for general network topologies and stubbornness values.

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