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Structural and Positional Encodings can significantly improve the performanceof Graph Neural Networks in downstream tasks. Recent literature has begun tosystematically investigate differences in the structural properties that theseapproaches encode, as well as performance trade-offs between them. However, thequestion of which structural properties yield the most effective encodingremains open. In this paper, we investigate this question from a geometricperspective. We propose a novel structural encoding based on discrete Riccicurvature (Local Curvature Profiles, short LCP) and show that it significantlyoutperforms existing encoding approaches. We further show that combining localstructural encodings, such as LCP, with global positional encodings improvesdownstream performance, suggesting that they capture complementary geometricinformation. Finally, we compare different encoding types with(curvature-based) rewiring techniques. Rewiring has recently received a surgeof interest due to its ability to improve the performance of Graph NeuralNetworks by mitigating over-smoothing and over-squashing effects. Our resultssuggest that utilizing curvature information for structural encodings deliverssignificantly larger performance increases than rewiring.

Effective Structural Encodings via Local Curvature Profiles