Robustness of large-scale stochastic matrices to localized perturbations

TL;DR

This paper provides bounds on how much the invariant distribution of a stochastic matrix can change due to localized perturbations, with applications to large-scale networks and algorithms.

Contribution

It introduces new bounds based on mixing and hitting times, improving understanding of robustness in large stochastic systems under localized changes.

Findings

Bounds depend on mixing time, hitting time, and escape time.

Results are useful even if the matrix difference is large in norm.

Applications include PageRank robustness and consensus algorithms.

Abstract

Upper bounds are derived on the total variation distance between the invariant distributions of two stochastic matrices differing on a subset W of rows. Such bounds depend on three parameters: the mixing time and the minimal expected hitting time on W for the Markov chain associated to one of the matrices; and the escape time from W for the Markov chain associated to the other matrix. These results, obtained through coupling techniques, prove particularly useful in scenarios where W is a small subset of the state space, even if the difference between the two matrices is not small in any norm. Several applications to large-scale network problems are discussed, including robustness of Google's PageRank algorithm, distributed averaging and consensus algorithms, and interacting particle systems.