Distributed Average Consensus in Sensor Networks with Random Link Failures

Abstract

We study the impact of the topology of a sensor network on distributed average consensus algorithms when the network links fail at random. We derive convergence results. In particular, we determine a sufficient condition for mean-square convergence of the distributed average consensus algorithm in terms of a moment of the distribution of the norm of a function of the network graph Laplacian matrix L (which is a random matrix, because the network links are random.) Further, because the computation of this moment involves costly simulations, we relate the mean-square convergence to the second eigenvalue of the mean Laplacian matrix, λ2(L), which is much easier to compute. We derive bounds on the convergence rate of the algorithm, which show that both the expected algebraic connectivity of the network, E[λ2(L)], and λ2(L) play an important role in determining the actual convergence rate. Specifically, larger values of E[λ2(L)] or λ2(L) lead to better convergence rates. Finally, we provide numerical studies that verify the analytical results.