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.

本文介绍了一种通过离散图Ricci曲率来增强图神经网络（GNN）的方法，并验证了在GNN中引入曲率信息可以缓解过度平滑等计算问题，并且基于曲率的图边丢弃算法进一步提高了模型对异质图的适应性，在同质图和异质图数据集上均优于现有基准模型，证明了在GNN中引入图几何信息的有效性。

曲率如何增强框架图卷积神经网络的适应能力