On Convergence Rate of a Continuous-Time Distributed Self-Appraisal Model with Time-Varying Relative Interaction Matrices

TL;DR

This paper analyzes the convergence behavior of a continuous-time social-confidence model with time-varying interactions, showing conditions for convergence to a democratic state and providing explicit convergence rates.

Contribution

It establishes convergence conditions for the model with doubly stochastic matrices and derives explicit exponential convergence rates.

Findings

Doubly stochastic matrices lead to convergence to equal confidence levels.

Non-doubly stochastic matrices may prevent convergence.

Explicit exponential convergence rate is derived.

Abstract

This paper studies a recently proposed continuous-time distributed self-appraisal model with time-varying interactions among a network of $n$ individuals which are characterized by a sequence of time-varying relative interaction matrices. The model describes the evolution of the social-confidence levels of the individuals via a reflected appraisal mechanism in real time. We first show by example that when the relative interaction matrices are stochastic (not doubly stochastic), the social-confidence levels of the individuals may not converge to a steady state. We then show that when the relative interaction matrices are doubly stochastic, the $n$ individuals' self-confidence levels will all converge to $1/n$, which indicates a democratic state, exponentially fast under appropriate assumptions, and provide an explicit expression of the convergence rate.