Distributed consensus with link failures as a structured stochastic uncertainty problem

We consider the standard distributed average consensus algorithm under the conditions of random communication link failures, for which we analyze convergence of nodes to average consensus in the mean square sense. We first recast this problem as a discrete-time linear system with multiplicative random coefficients. We then rewrite the system equations as a nominal system in feedback with diagonally structured time-varying stochastic uncertainty; a problem for which necessary and sufficient mean square stability conditions have recently been derived. We investigate the particular instance of these conditions in the case of networked consensus with random link failures. In particular, we show that for circulant graphs, mean square convergence is guaranteed for any probability of link failure other than 1. We anticipate our particular analysis techniques to be applicable to the robust performance problem as well.