Theoretically and Practically Efficient Resistance Distance Computation on Large Graphs

The computation of resistance distance is pivotal in a wide range of graph analysis applications, including graph clustering, link prediction, and graph neural networks. Despite its foundational importance, efficient algorithms for computing resistance distances on large graphs are still lacking. Existing state-of-the-art (SOTA) methods, including power iteration-based algorithms and random walk-based local approaches, often struggle with slow convergence rates, particularly when the condition number of the graph Laplacian matrix, denoted by $\kappa$, is large. To tackle this challenge, we propose two novel and efficient algorithms inspired by the classic Lanczos method: Lanczos Iteration and Lanczos Push, both designed to reduce dependence on $\kappa$. Among them, Lanczos Iteration is a near-linear time global algorithm, whereas Lanczos Push is a local algorithm with a time complexity independent of the size of the graph. More specifically, we prove that the time complexity of Lanczos Iteration is $\tilde{O}(\sqrt{\kappa} m)$ ($m$ is the number of edges of the graph and $\tilde{O}$ means the complexity omitting the $\log$ terms) which achieves a speedup of $\sqrt{\kappa}$ compared to previous power iteration-based global methods. For Lanczos Push, we demonstrate that its time complexity is $\tilde{O}(\kappa^{2.75})$ under certain mild and frequently established assumptions, which represents a significant improvement of $\kappa^{0.25}$ over the SOTA random walk-based local algorithms. We validate our algorithms through extensive experiments on eight real-world datasets of varying sizes and statistical properties, demonstrating that Lanczos Iteration and Lanczos Push significantly outperform SOTA methods in terms of both efficiency and accuracy.