Computable Convergence Rate Bound for Ratio Consensus Algorithms

The objective of this letter 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.
from 2013 on similar bounds that have been obtained in a more restrictive setup with limited conclusions. The present paper complements results by Gerencsér and Gerencsér from 2022, 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 this letter will be compared to the actual rate of almost sure convergence experimentally on a range of modulated random geometric graphs with random local interactions.