Beyond Distance: Reliability-Gated Fisher–Rao Routing for Heterophilic Graphs

Heterophilic graphs connect nodes with different labels, yet nodes of the same class can retain characteristic neighborhood label distributions (NLDs). Existing methods primarily treat NLDs as Euclidean statistics, auxiliary targets, or pairwise similarity signals. We instead place NLDs on the probability simplex and distinguish the radial magnitude of a Fisher–Rao displacement from its tangent direction. This motivates FR-NLD, a reliability-gated geometric network in which every neighbor acts as a viewer of the target node and candidate class prototype. The model compares the target and prototype in the Fisher tangent space anchored at the neighbor, using angular evidence only when both displacements are sufficiently reliable and otherwise falling back to radial evidence. Degree-aware NLD estimation regularizes finite neighborhoods, while residual routing and belief refresh support repeated geometric inference. Experiments on five modern heterophily benchmarks using all ten public splits show that FR-NLD achieves the highest Minesweeper score among the recent published comparators in our evaluation and remains competitive on Roman-empire and Tolokers. On Minesweeper, it outperforms reproduced Polynormer while using only 21% of its parameters and peak GPU memory. Controlled checkpoint interventions further show that predictions depend on the neighbor-specific tangent reference and angular evidence, supporting the proposed geometric mechanism.