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.

翻译：结构和位置编码可显著提升图神经网络在下游任务中的性能。近期文献已开始系统研究这些编码方法所编码的结构属性差异及其性能权衡。然而，哪些结构属性能产生最有效的编码仍是一个开放性问题。本文从几何视角探究该问题，提出一种基于离散里奇曲率的新型结构编码（局部曲率剖面，简称LCP），并证明其显著优于现有编码方法。我们进一步表明，将局部结构编码（如LCP）与全局位置编码相结合可提升下游性能，说明它们捕获了互补的几何信息。最后，我们比较了不同编码类型与（基于曲率的）重连技术。重连因能通过缓解过平滑和过挤压效应提升图神经网络性能而近期备受关注。我们的结果表明，与重连相比，利用曲率信息进行结构编码能带来更显著的性能提升。