On Algorithmic Robustness of Corrupted Markov Chains

2025-07-20

AI Summary

This paper investigates the robustness of stationary distributions of finite Markov chains under adversarial corruption of the transition matrix. We propose a robustification method grounded in the PageRank mechanism: assuming only mild continuity of the restart distribution, we employ spectral gap analysis and total variation distance to establish dimension-independent stability—namely, for any arbitrarily small fraction ε of corrupted transition edges, the perturbed PageRank distribution deviates from the original by at most poly(ε). This result is the first to reveal that PageRank itself serves as an intrinsic, parameter-free regularizer, requiring no additional hyperparameter tuning. It provides both theoretical foundations and practical guarantees for designing robust graph learning and ranking algorithms under adversarial structural perturbations.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Computer Vision: Adversarial Attacks & Robustness
• Natural Language Processing: Safety and Robustness

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 $ε$-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.

Problem

• Analyzing robustness of Markov chains to adversarial corruptions
• Studying PageRank's resilience against edge corruptions
• Ensuring dimension-free algorithmic guarantees for corrupted chains

Innovation

• PageRank variants ensure robustness against corruptions
• Spectral gap Markov chains resist adversarial changes
• Dimension-free guarantees for corrupted transition matrices