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.

翻译：判别随机游走（DRW）是半监督节点分类中一种简单而强大的工具，但其理论基础仍不完整。我们通过信息几何的视角重新审视DRW，将吸收马尔可夫链上类别特定的首达时律族视为一个统计流形。从对数线性边权模型出发，我们推导了首达时概率质量函数、其完整矩层次结构以及观测Fisher信息的闭式表达式。每个种子节点的Fisher矩阵均为秩一矩阵，通过对其零空间取商，我们得到了一个低维、全局平坦的流形，该流形捕捉了模型所有可识别方向。利用该几何结构，我们为未标记节点引入了一个敏感度评分，该评分界定了单位Fisher扰动下DRW介数的一阶最大变化，在一维情况下可达该上界。该评分可引导出主动标签获取、边重加权及解释的基于原理的策略。