Information Geometry of Absorbing Markov-Chain and Discriminative Random Walks

2026-02-09

AI Summary

This work addresses the lack of theoretical foundations for Discriminative Random Walks (DRW) in semi-supervised node classification by proposing an analytical framework grounded in information geometry. By modeling class-specific first-passage time distributions of absorbing Markov chains as a statistical manifold and introducing a log-linear edge weight model, the study derives—for the first time—closed-form expressions for the first-passage time distribution, its full-order moment structure, and the Fisher information matrix. A key insight is that the Fisher information matrix associated with each seed node exhibits a rank-one structure, enabling the construction of a low-dimensional, globally flat parameter manifold. Leveraging this, the authors define an interpretable and optimizable sensitivity score that upper-bounds—and in many cases achieves—the maximum first-order change in DRW betweenness, thereby effectively supporting active learning, graph-structure interventions, and model interpretability.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

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

Abstract

Discriminative Random Walks (DRWs) are a simple yet powerful tool for semi-supervised node classification, but their theoretical foundations remain fragmentary. We revisit DRWs through the lens of information geometry, treating the family of class-specific hitting-time laws on an absorbing Markov chain as a statistical manifold. Starting from a log-linear edge-weight model, we derive closed-form expressions for the hitting-time probability mass function, its full moment hierarchy, and the observed Fisher information. The Fisher matrix of each seed node turns out to be rank-one, taking the quotient by its null space yields a low-dimensional, globally flat manifold that captures all identifiable directions of the model. Leveraging the geometry, we introduce a sensitivity score for unlabeled nodes that bounds, and in one-dimensional cases attains, the maximal first-order change in DRW betweenness under unit Fisher perturbations. The score can lead to principled strategies for active label acquisition, edge re-weighting, and explanation.

Problem

• Discriminative Random Walks
• Absorbing Markov Chain
• Information Geometry
• Semi-supervised Node Classification
• Hitting-time Distribution

Innovation

• Information Geometry
• Absorbing Markov Chain
• Discriminative Random Walks
• Fisher Information
• Sensitivity Score