Information Geometry of Absorbing Markov-Chain and Discriminative Random Walks

TL;DR

This paper applies information geometry to analyze Discriminative Random Walks on absorbing Markov chains, deriving explicit formulas and a sensitivity score to improve semi-supervised node classification and active learning strategies.

Contribution

It introduces a geometric framework for DRWs, deriving closed-form expressions and a sensitivity score for better understanding and improving semi-supervised learning.

Findings

Fisher matrix of seed nodes is rank-one.

Derived closed-form hitting-time probability and moments.

Introduced a sensitivity score for active learning and explanation.

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.