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 multi-agent system with a time-varying network, where the associated stochastic matrices have a time-independent Perron vector. This method provides bounds on convergence rates that unify and refine most of the previously known bounds. They depend on geometric parameters of the dynamic communication graph such as the normalized diameter or the bottleneck measure. As corollaries of these geometric bounds, we show that the convergence rate of the Metropolis algorithm in a system of $n$ agents is less than $1-1/4n^2$ with any communication graph that may vary in time, but 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 that the communication graph is not too irregular. Moreover our bounds offer improved convergence rates for several averaging algorithms and specific families of communication graphs. Finally we extend our methodology to a time-varying Perron vector and show how convergence times may dramatically degrade with even limited variations of Perron vectors.