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

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.