The Role of Edge Geometry in Graph Neural Networks

While much work on graph neural networks has focused on designing node features, some structural information, such as curvature, is naturally defined on edges. Such information can help in two ways: by supplying structural information that standard message passing does not recover, or by improving message propagation. We separate the two and ask when each helps, and for which architectures and tasks. We introduce curvature profiles that combine degree-based curvature and counts of cycles of different lengths. These profiles can be used as edge features in standard graph networks or to guide message propagation in our curvature-aware RicciMP layer. Exact constructions show how profiles can improve filtering and how neighboring context reveals curvature information absent from one edge's profile. At the same time, longer cycles yield a strict expressivity hierarchy: for each fixed $k\ge 3$ curvature-profile refinement is incomparable with the coordinate-wise $k$-Weisfeiler-Leman test once the cycle order reaches an explicit $k$-dependent threshold, each distinguishing graph pairs the other misses. Because profiles are computed before any message passing, this reach is obtained by changing the labels refinement reads, not the refinement procedure. Empirically, we show that whether curvature is most useful as a feature or for propagation depends on the task. In particular, curvature profiles substantially improve molecular prediction with GINE, achieving scores close to matched subgraph-counting models, and benefit baseline classification benchmarks and community-recovery tasks. RicciMP also improves image-region labeling over GINE with the same structural features and unweighted multi-hop propagation. On long-range tasks, augmentation effects are mixed, and RicciMP trails the strongest tested baselines without curvature profiles. Together, these results distinguish when structural augmentation helps prediction and what the tested propagation choices add.