Graph Neural Networks (GNNs) had been demonstrated to be inherently susceptible to the problems of over-smoothing and over-squashing. These issues prohibit the ability of GNNs to model complex graph interactions by limiting their effectiveness in taking into account distant information. Our study reveals the key connection between the local graph geometry and the occurrence of both of these issues, thereby providing a unified framework for studying them at a local scale using the Ollivier-Ricci curvature. Specifically, we demonstrate that over-smoothing is linked to positive graph curvature, while over-squashing is linked to negative graph curvature. Based on our theory, we propose the Batch Ollivier-Ricci Flow, a novel rewiring algorithm capable of simultaneously addressing both over-smoothing and over-squashing.

翻译：图神经网络（GNN）已被证明固有地易受过平滑与过挤压问题的影响。这些问题通过限制GNN有效整合远距离信息的能力，阻碍其对复杂图交互的建模。本研究揭示了局部图几何结构与这两类问题发生之间的关键联系，从而为在局部尺度上利用奥利维耶-里奇曲率统一研究这些问题提供了理论框架。具体而言，我们证明过平滑与正图曲率相关，而过挤压与负图曲率相关。基于这一理论，我们提出了批奥利维耶-里奇流（Batch Ollivier-Ricci Flow），这是一种能够同时解决过平滑与过挤压问题的全新图重连算法。