Geometric bounds for convergence rates of averaging algorithms

We develop a generic method for bounding the convergence rate of an averaging algorithm running in a multiagent system with a time-varying network, where the associated stochastic matrices have a time-independent Perron vector. The resulting bounds depend on geometric parameters of the dynamic communication graph such as the weighted diameter or the bottleneck measure. As corollaries, we show that the convergence rate of the Metropolis algorithm in a system of n agents is less than 1 - 1/4n2 if the communication graph is permanently connected and bidirectional. We prove a similar upper bound for the EqualNeighbor algorithm under the additional assumptions that the number of neighbors of each agent is constant and the communication graph is not too irregular. In general, our bounds offer improved convergence rates for several averaging algorithms and specific families of communication graphs.