On performance of consensus protocols subject to noise: role of hitting times and network structure

We study the performance of linear consensus protocols based on repeated averaging in the presence of additive noise. When the consensus dynamics corresponds to a reversible Markov chain, we give an exact expression for the weighted steady-state disagreement in terms of the stationary distribution and hitting times in an underlying graph. We show how this expression can be used to characterize the asymptotic mean-square disagreement in certain noisy opinion dynamics, as well as the scalability of protocols for formation control and decentralized clock synchronization.