Distributed Evaluation and Convergence of Self-Appraisals in Social Networks

TL;DR

This paper models opinion dynamics in social networks where agents iteratively evaluate their self-appraisals, and demonstrates that these evaluations converge exponentially to a stable equilibrium in a distributed manner.

Contribution

It introduces a novel distributed dynamical system for self-appraisal evaluation in social networks with rooted digraph topology, proving convergence to equilibrium.

Findings

Self-appraisals converge exponentially to a stable equilibrium.

The model operates in continuous time with distributed communication.

Convergence holds for almost all initial conditions.

Abstract

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.