Finite Rate Distributed Weight-Balancing and Average Consensus Over Digraphs

This article proposes the first distributed algorithm that solves the weight-balancing problem using only finite rate and simplex communications among nodes, compliant with the directed nature of the graph edges. It is proved that the algorithm converges to a weight-balanced solution at sublinear rate. The analysis builds upon a new metric inspired by positional system representations, which characterizes the dynamics of information exchange over the network, and on a novel step-size rule. Building on this result, a novel distributed algorithm is proposed that solves the
average
consensus problem over digraphs, using, at each timeslot, finite rate simplex communications between adjacent nodes—some bits for the weight-balancing problem and others for the average consensus. Convergence of the proposed quantized consensus algorithm to the average of the node's
unquantized
initial values is established, both almost surely and in the moment generating function of the error; and a sublinear convergence rate is proved for sufficiently large step-sizes. Numerical results validate our theoretical findings.