On Algorithmic Robustness of Corrupted Markov Chains

TL;DR

This paper investigates the robustness of PageRank-based Markov chains against adversarial transition matrix corruptions, demonstrating that PageRank maintains a close approximation to the original stationary distribution under broad, realistic conditions.

Contribution

The paper establishes that PageRank chains are robust to adversarial edge corruptions in Markov chains with spectral gaps, providing dimension-free guarantees.

Findings

PageRank distributions are stable under adversarial corruptions.

Robustness guarantees scale gracefully with natural parameters.

PageRank acts as a regularizer against broad corruptions.

Abstract

We study the algorithmic robustness of general finite Markov chains in terms of their stationary distributions to general, adversarial corruptions of the transition matrix. We show that for Markov chains admitting a spectral gap, variants of the \emph{PageRank} chain are robust in the sense that, given an \emph{arbitrary} corruption of the edges emanating from an $\epsilon$-measure of the nodes, the PageRank distribution of the corrupted chain will be $\mathsf{poly}(\varepsilon)$ close in total variation to the original distribution under mild conditions on the restart distribution. Our work thus shows that PageRank serves as a simple regularizer against broad, realistic corruptions with algorithmic guarantees that are dimension-free and scale gracefully in terms of necessary and natural parameters.