Computable convergence rate bound for ratio consensus algorithms

TL;DR

This paper derives a computable upper bound for the almost sure convergence rate of ratio consensus algorithms, extending previous theoretical results and validating the bounds through experiments on random graph models.

Contribution

It provides a new, computable upper bound for the convergence rate of ratio consensus algorithms, improving upon prior restrictive bounds and complementing existing theoretical results.

Findings

The upper bound closely matches the actual convergence rate in experiments.

The bound is applicable to a broad class of ratio consensus algorithms.

Experimental validation on random graphs confirms the theoretical predictions.

Abstract

The objective of the paper is to establish a computable upper bound for the almost sure convergence rate for a class of ratio consensus algorithms defined via column-stochastic matrices. Our result extends the works of Iutzeler et al. (2013) on similar bounds that have been obtained in a more restrictive setup with limited conclusions. The present paper complements the results of Gerencs\'er and Gerencs\'er (2021), identifying the exact almost sure convergence rate of a wide class of ratio consensus algorithms in terms of a spectral gap, which is, however, not computable in general. The upper bound provided in the paper will be compared to the actual rate of almost sure convergence experimentally on a range of modulated random geographic graphs with random local interactions.