ASEHybrid: When Geometry Matters Beyond Homophily in Graph Neural Networks
Shalima Binta Manir, Tim Oates
TL;DR
The paper tackles when geometry-aware GNNs outperform feature-only baselines by reframing graph usefulness through label informativeness (LI) and the conditional edge–label information $I(Y;E\mid \tilde{X})$. It develops a unified theory that links curvature-guided diffusion, Laplacian positional encodings, and curvature-guided rewiring, proving a necessary-and-sufficient condition for geometry to help and showing that local curvature alone cannot boost expressivity beyond 1-WL. The practical ASEHybrid architecture embodies these ideas by fusing curvature summaries, global spectral encodings, and curvature-conditioned attention, with a curtailable rewiring component that converges and remains stable under bounded edits. Empirically, geometry-aware enhancements yield substantial gains on label-informative heterophilous graphs (e.g., Chameleon, Squirrel, Texas) and modest or no gains in low-LI or high-baseline regimes, validating the proposed information-theoretic criterion and guiding when to deploy curvature and topology adaptations. Overall, the work provides both rigorous guarantees and actionable design principles for geometry-aware GNNs in heterogeneous network settings, with implications for robust diffusion and improved label propagation where graph structure carries label-relevant information beyond node features.
Abstract
Standard message-passing graph neural networks (GNNs) often struggle on graphs with low homophily, yet homophily alone does not explain this behavior, as graphs with similar homophily levels can exhibit markedly different performance and some heterophilous graphs remain easy for vanilla GCNs. Recent work suggests that label informativeness (LI), the mutual information between labels of adjacent nodes, provides a more faithful characterization of when graph structure is useful. In this work, we develop a unified theoretical framework that connects curvature-guided rewiring and positional geometry through the lens of label informativeness, and instantiate it in a practical geometry-aware architecture, ASEHybrid. Our analysis provides a necessary-and-sufficient characterization of when geometry-aware GNNs can improve over feature-only baselines: such gains are possible if and only if graph structure carries label-relevant information beyond node features. Theoretically, we relate adjusted homophily and label informativeness to the spectral behavior of label signals under Laplacian smoothing, show that degree-based Forman curvature does not increase expressivity beyond the one-dimensional Weisfeiler--Lehman test but instead reshapes information flow, and establish convergence and Lipschitz stability guarantees for a curvature-guided rewiring process. Empirically, we instantiate ASEHybrid using Forman curvature and Laplacian positional encodings and conduct controlled ablations on Chameleon, Squirrel, Texas, Tolokers, and Minesweeper, observing gains precisely on label-informative heterophilous benchmarks where graph structure provides label-relevant information beyond node features, and no meaningful improvement in high-baseline regimes.