Opinion Dynamics in Heterogeneous Networks: Convergence Conjectures and Theorems

Recently, significant attention has been dedicated to the models of opinion dynamics in which opinions are described by real numbers, and agents update their opinions synchronously by averaging their neighbors' opinions. The neighbors of each agent can be defined as either (1) those agents whose opinions are in its "confidence range," or (2) those agents whose "influence range" contain the agent's opinion. The former definition is employed in Hegselmann and Krause's bounded confidence model, and the latter is novel here. As the confidence and influence ranges are distinct for each agent, the heterogeneous state-dependent interconnection topology leads to a poorly-understood complex dynamic behavior. In both models, we classify the agents via their interconnection topology and, accordingly, compute the equilibria of the system. Then, we define a positive invariant set centered at each equilibrium opinion vector. We show that if a trajectory enters one such set, then it converges to a steady state with constant interconnection topology. This result gives us a novel sufficient condition for both models to establish convergence, and is consistent with our conjecture that all trajectories of the bounded confidence and influence models eventually converge to a steady state under fixed topology.Comment: 22 pages, Submitted to SIAM Journal on Control and Optimization (SICON

On the reflected appraisals dynamics of influence networks with stubborn agents

Abstract — This article focuses on the evolution of interper-sonal 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 self-weight 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. I

More than a schoolgirl crush: Amy Adler and the adolescent fan

A schoolgirl crush: a mixture of desire, identification and aggression, played out in fantasy. A combination of narcissistic, heterosexual and homosexual desire, the intense identification with the object of desire undoes the paradigms of ‘mature’ heterosexuality. In the work of American artist Amy Adler , this adolescent position is appropriated and reworked in her hybrid photographed drawings. Depicting a range of adolescent and child characters, including the artist as a young woman, Adler’s portraits explore the ways in which identity is filtered through celebrity culture, with her own images becoming part of a seductive, androgynous cast that include a young River Phoenix and a series of unnamed, nubile female stars taken from magazines, billboards and CD covers. In this work Adler enacts the adolescent fan copying a photograph of his or her idol as perfectly as possible, an act of ownership that inscribes the fan’s desire into the image. However, as Adler re-enacts this process, she maintains a distance, the adult performing the adolescent who attempts to possess or perhaps become the object of desire through the act of drawing. This paper explores the potential of the adolescent position for thinking through modes of desiring and identification that are often dismissed as ‘immature’ or ‘feminine’, undermining the stability of binary definitions of both sexuality and gender. In thinking through the aggression and seduction presented in Adler’s work, the adolescent fan provides a structure for thinking about female desire that allows it to be more than a schoolgirl crush

Opinion Dynamics and the Evolution of Social Power in Influence Networks

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). © 2015 Society for Industrial and Applied Mathematics