Ergodicity of Continuous-Time Distributed Averaging Dynamics: A Spanning Directed Rooted Tree Approach

In this article, we consider time-varying distributed averaging dynamics. Motivated by a necessary condition on the ergodicity, we provide a sufficient condition for the ergodicity of such dynamics. We show that if groups of agents are connected using a directed acyclic graph containing a spanning directed rooted tree and the averaging dynamics restricted to each group is
$\mathcal {P}^*$
, then the dynamics over the whole network is ergodic. In particular, this provides a general condition for convergence of consensus dynamics where

groups

of agents capable of reaching consensus follow each other on a time-varying network.