Efficient Curvature-aware Graph Network
Chaoqun Fei, Tinglve Zhou, Tianyong Hao, Yangyang Li
TL;DR
This work tackles the scalability bottleneck of curvature-aware graph learning by introducing Effective Resistance Curvature (ERC), a curvature measure derived from the graph Laplacian’s effective resistance rather than optimal transport. ERC retains the geometric expressiveness of Ollivier-Ricci curvature while dramatically reducing computational cost through stable Laplacian inversions and perturbation-based approximations. Theoretical results establish stability and a close relationship to OR curvature, and extensive experiments show ERC achieves performance close to OR with orders-of-magnitude speedups, including in graph representation learning and graph pooling. This makes curvature-aware GNNs practical for large-scale graphs and provides a robust, scalable alternative to Wasserstein-based curvature methods.
Abstract
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.