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

Evolution of Social Power for Opinion Dynamics Networks

This article studies the evolution of opinions and interpersonal influence structures in a group of agents as they discuss a sequence of issues, each of which follows an opinion dynamics model. In this work, we propose a general opinion dynamics model and an evolution of interpersonal influence structures based on the model of reflected appraisals proposed by Friedkin. Our contributions can be summarized as follows: (i) we introduce a model of opinion dynamics and evolution of interpersonal influence structures between issues viewed as a best response cost minimization to the neighbor's actions, (ii) we show that DeGroot's and Friedkin-Johnsen's models of opinion dynamics and their evolution of interpersonal influence structures are particular cases of our proposed model, and (iii) we prove the existence of an equilibrium. This work is a step towards providing a solid formulation of the evolution of opinions and interpersonal influence structures over a sequence of issues

On a Modified DeGroot-Friedkin Model of Opinion Dynamics

This paper studies the opinion dynamics that result when individuals consecutively discuss a sequence of issues. Specifically, we study how individuals' self-confidence levels evolve via a reflected appraisal mechanism. Motivated by the DeGroot-Friedkin model, we propose a Modified DeGroot-Friedkin model which allows individuals to update their self-confidence levels by only interacting with their neighbors and in particular, the modified model allows the update of self-confidence levels to take place in finite time without waiting for the opinion process to reach a consensus on any particular issue. We study properties of this Modified DeGroot-Friedkin model and compare the associated equilibria and stability with those of the original DeGroot-Friedkin model. Specifically, for the case when the interaction matrix is doubly stochastic, we show that for the modified model, the vector of individuals' self-confidence levels asymptotically converges to a unique nontrivial equilibrium which for each individual is equal to 1/n, where n is the number of individuals. This implies that eventually, individuals reach a democratic state

Distributed Evaluation and Convergence of Self-Appraisals in Social Networks

We consider in this paper a networked system of opinion dynamics in continuous time, where the agents are able to evaluate their self-appraisals in a distributed way. In the model we formulate, the underlying network topology is described by a rooted digraph. For each ordered pair of agents $(i,j)$, we assign a function of self-appraisal to agent $i$, which measures the level of importance of agent $i$ to agent $j$. Thus, by communicating only with her neighbors, each agent is able to calculate the difference between her level of importance to others and others' level of importance to her. The dynamical system of self-appraisals is then designed to drive these differences to zero. We show that for almost all initial conditions, the trajectory generated by this dynamical system asymptotically converges to an equilibrium point which is exponentially stable

Dynamical Networks of Social Influence: Modern Trends and Perspectives

Dynamics and control of processes over social networks, such as the evolution of opinions, social influence and interpersonal appraisals, diffusion of information and misinformation, emergence and dissociation of communities, are now attracting significant attention from the broad research community that works on systems, control, identification and learning. To provide an introduction to this rapidly developing area, a Tutorial Session was included into the program of IFAC World Congress 2020. This paper provides a brief summary of the three tutorial lectures, covering the most “mature” directions in analysis of social networks and dynamics over them: 1) formation of opinions under social influence; 2) identification and learning for analysis of a network’s structure; 3) dynamics of interpersonal appraisals

Distributed Learning from Interactions in Social Networks

We consider a network scenario in which agents can evaluate each other according to a score graph that models some interactions. The goal is to design a distributed protocol, run by the agents, that allows them to learn their unknown state among a finite set of possible values. We propose a Bayesian framework in which scores and states are associated to probabilistic events with unknown parameters and hyperparameters, respectively. We show that each agent can learn its state by means of a local Bayesian classifier and a (centralized) Maximum-Likelihood (ML) estimator of parameter-hyperparameter that combines plain ML and Empirical Bayes approaches. By using tools from graphical models, which allow us to gain insight on conditional dependencies of scores and states, we provide a relaxed probabilistic model that ultimately leads to a parameter-hyperparameter estimator amenable to distributed computation. To highlight the appropriateness of the proposed relaxation, we demonstrate the distributed estimators on a social interaction set-up for user profiling.Comment: This submission is a shorter work (for conference publication) of a more comprehensive paper, already submitted as arXiv:1706.04081 (under review for journal publication). In this short submission only one social set-up is considered and only one of the relaxed estimators is proposed. Moreover, the exhaustive analysis, carried out in the longer manuscript, is completely missing in this versio