  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.
