Convergence Rate Analysis of Ratio Consensus Algorithms with Column-Allowable Matrices

We give almost sure convergence rate bounds of ratio consensus algorithms when the protocol can be reformulated to be linear updates of vector values on a possibly larger, augmented network. This is an improvement of the results of Gerencs\'er and Gerencs\'er from 2021 by allowing zero values on auxiliary nodes infinitely often which makes the technique applicable to a much larger family of algorithms.