Limited Rate Distributed Weight-Balancing and Average Consensus Over Digraphs

Abstract: Distributed quantized weight-balancing and average consensus over fixed digraphs are considered. A digraph with non-negative weights associated to its edges is weight-balanced if, for each node, the sum of the weights of its out-going edges is equal to that of its incoming edges. This paper proposes and analyzes the first distributed algorithm that solves the weight-balancing problem using only finite rate and simplex communications among nodes (compliant to the directed nature of the graph edges). Asymptotic convergence of the scheme is proved and a convergence rate analysis is provided. Building on this result, a novel distributed algorithm is proposed that solves the average consensus problem over digraphs, using, at each iteration, finite rate simplex communications between adjacent nodes -- some bits for the weight-balancing problem, other for the average consensus. Convergence of the proposed quantized consensus algorithm to the average of the real (i.e., unquantized) agent's initial values is proved, both almost surely and in $r$th mean for all positive integer $r$. Finally, numerical results validate our theoretical findings.