Finite Rate Distributed Weight-Balancing and Average Consensus Over Digraphs

Chang-Shen Lee, Nicolò Michelusi, Gesualdo Scutari

Abstract: 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">average</i> 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">unquantized</i> 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.