We study the problem of efficiently approximating the effective resistance (ER) on undirected graphs, where ER is a widely used node proximity measure with applications in graph spectral sparsification, multi-class graph clustering, network robustness analysis, graph machine learning, and more. Specifically, given any nodes s and t in an undirected graph G, we aim to efficiently estimate the ER value R(s,t) between nodes s and t, ensuring a small absolute error ε. The previous best algorithm for this problem has a worst-case computational complexity of Õ(Lmax3/ε2d2), where the value of Lmax depends on the mixing time of random walks on G, d = mind(s), d(t), and d(s), d(t) denote the degrees of nodes s and t, respectively. We improve this complexity to Õ ( min Lmax7/3/ε2/3, Lmax3/ε2d2,mLmax ), achieving a theoretical improvement of Õ (maxLmax2/3/ε4/3d2,1, Lmax2/ε2d2m2) over previous results. Here, m denotes the number of edges. Given that Lmax is often very large in real-world networks (e.g., Lmax > 104), our improvement on Lmax is significant, especially for real-world networks. We also conduct extensive experiments on real-world and synthetic graph datasets to empirically demonstrate the superiority of our method. The experimental results show that our method achieves a 10× to 1000× speedup in running time while maintaining the same absolute error compared to baseline methods.

Graph Neural Networks (GNNs) have emerged as a prominent research topic in the field of machine learning. Existing GNN models are commonly categorized into two types: spectral GNNs, which are designed based on polynomial graph filters, and spatial GNNs, which utilize a message-passing scheme as the foundation of the model. For the expressive power and universality of spectral GNNs, a natural approach is to improve the design of basis functions for better approximation ability. As for spatial GNNs, models like Graph Isomorphism Networks (GIN) analyze their expressive power based on Graph Isomorphism Tests. Recently, there have been attempts to establish connections between spatial GNNs and geometric concepts like curvature and cellular sheaves, as well as physical phenomena like oscillators. However, despite the recent progress, there is still a lack of comprehensive analysis regarding the universality of spatial GNNs from the perspectives of geometry and physics.