Graph neural network (GNN) has been demonstrated powerful in modeling graph-structured data. However, despite many successful cases of applying GNNs to various graph classification and prediction tasks, whether the graph geometrical information has been fully exploited to enhance the learning performance of GNNs is not yet well understood. This paper introduces a new approach to enhance GNN by discrete graph Ricci curvature. Specifically, the graph Ricci curvature defined on the edges of a graph measures how difficult the information transits on one edge from one node to another based on their neighborhoods. Motivated by the geometric analogy of Ricci curvature in the graph setting, we prove that by inserting the curvature information with different carefully designed transformation function $\zeta$, several known computational issues in GNN such as over-smoothing can be alleviated in our proposed model. Furthermore, we verified that edges with very positive Ricci curvature (i.e., $\kappa_{i,j} \approx 1$) are preferred to be dropped to enhance model's adaption to heterophily graph and one curvature based graph edge drop algorithm is proposed. Comprehensive experiments show that our curvature-based GNN model outperforms the state-of-the-art baselines in both homophily and heterophily graph datasets, indicating the effectiveness of involving graph geometric information in GNNs.

翻译：图神经网络（GNN）已被证明在处理图结构数据方面具有强大能力。然而，尽管GNN在许多图分类和预测任务中取得了许多成功案例，但图几何信息是否已被充分利用以提升GNN的学习性能尚未得到充分理解。本文提出了一种利用离散图里奇曲率增强GNN的新方法。具体而言，定义在图边上的里奇曲率根据邻居结构衡量了信息沿边从一个节点传递到另一个节点的难度。受图设置中里奇曲率的几何类比启发，我们证明通过使用不同精心设计的变换函数ζ插入曲率信息，可以缓解我们提出的模型中GNN的一些已知计算问题（例如过平滑）。此外，我们验证了具有非常正里奇曲率（即κ_{i,j} ≈ 1）的边更倾向于被丢弃，以增强模型对异质性图的适应能力，并提出了一种基于曲率的图边丢弃算法。大量实验表明，我们的基于曲率的GNN模型在同质性和异质性图数据集上均优于最先进的基线方法，这表明在图神经网络中引入图几何信息的有效性。