  Structural and Positional Encodings can significantly improve the performance
of Graph Neural Networks in downstream tasks. Recent literature has begun to
systematically investigate differences in the structural properties that these
approaches encode, as well as performance trade-offs between them. However, the
question of which structural properties yield the most effective encoding
remains open. In this paper, we investigate this question from a geometric
perspective. We propose a novel structural encoding based on discrete Ricci
curvature (Local Curvature Profiles, short LCP) and show that it significantly
outperforms existing encoding approaches. We further show that combining local
structural encodings, such as LCP, with global positional encodings improves
downstream performance, suggesting that they capture complementary geometric
information. Finally, we compare different encoding types with
(curvature-based) rewiring techniques. Rewiring has recently received a surge
of interest due to its ability to improve the performance of Graph Neural
Networks by mitigating over-smoothing and over-squashing effects. Our results
suggest that utilizing curvature information for structural encodings delivers
significantly larger performance increases than rewiring.
