Graph curvature provides geometric priors for Graph Neural Networks (GNNs), enhancing their ability to model complex graph structures, particularly in terms of structural awareness, robustness, and theoretical interpretability. Among existing methods, Ollivier-Ricci curvature has been extensively studied due to its strong geometric interpretability, effectively characterizing the local geometric distribution between nodes. However, its prohibitively high computational complexity limits its applicability to large-scale graph datasets. To address this challenge, we propose a novel graph curvature measure--Effective Resistance Curvature--which quantifies the ease of message passing along graph edges using the effective resistance between node pairs, instead of the optimal transport distance. This method significantly outperforms Ollivier-Ricci curvature in computational efficiency while preserving comparable geometric expressiveness. Theoretically, we prove the low computational complexity of effective resistance curvature and establish its substitutability for Ollivier-Ricci curvature. Furthermore, extensive experiments on diverse GNN tasks demonstrate that our method achieves competitive performance with Ollivier-Ricci curvature while drastically reducing computational overhead.

翻译：图曲率为图神经网络（GNN）提供了几何先验，增强了其建模复杂图结构的能力，特别是在结构感知、鲁棒性和理论可解释性方面。在现有方法中，Ollivier-Ricci曲率因其强大的几何可解释性而被广泛研究，能有效刻画节点间的局部几何分布。然而，其极高的计算复杂度限制了其在大规模图数据集上的适用性。为应对这一挑战，我们提出了一种新颖的图曲率度量——有效电阻曲率，该方法利用节点对之间的有效电阻（而非最优传输距离）来量化沿图边进行消息传递的难易程度。此方法在计算效率上显著优于Ollivier-Ricci曲率，同时保持了相当的几何表达能力。理论上，我们证明了有效电阻曲率的低计算复杂度，并确立了其对Ollivier-Ricci曲率的可替代性。此外，在多种GNN任务上的大量实验表明，我们的方法在极大降低计算开销的同时，取得了与Ollivier-Ricci曲率相竞争的性能。