# Finite rate distributed weight-balancing and average consensus over digraphs

## TL;DR

This paper introduces a novel distributed algorithm for weight-balancing and average consensus over directed graphs, using finite rate simplex communications, with proven convergence and sublinear rate, validated by numerical experiments.

## Contribution

It presents the first finite rate distributed weight-balancing algorithm and extends it to average consensus, with new metrics and step-size rules for directed graphs.

## Key findings

- Algorithm converges to weight-balanced solution at sublinear rate
- Proposed consensus algorithm converges to true average almost surely
- Numerical results confirm theoretical convergence and performance

## Abstract

This paper 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.