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 $κ$, 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 $κ$. 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 $\tilde{O}(\sqrtκ m)$ ($m$ is the number of edges of the graph and $\tilde{O}$ means the complexity omitting the $\log$ terms) which achieves a speedup of $\sqrtκ$ compared to previous power iteration-based global methods. For Lanczos Push, we demonstrate that its time complexity is $\tilde{O}(κ^{2.75})$ under certain mild and frequently established assumptions, which represents a significant improvement of $κ^{0.25}$ 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.

翻译：电阻距离计算在图聚类、链接预测和图神经网络等广泛的图分析应用中具有关键作用。尽管其基础重要性显著，但针对大规模图的高效电阻距离计算算法仍然缺乏。现有的最先进方法，包括基于幂迭代的算法和基于随机游走的局部方法，通常面临收敛速度缓慢的问题，尤其是在图拉普拉斯矩阵的条件数（记为$κ$）较大时。为应对这一挑战，我们受经典Lanczos方法启发，提出了两种新颖高效算法：Lanczos迭代和Lanczos推送，两者均旨在降低对$κ$的依赖。其中，Lanczos迭代是一种近线性时间的全局算法，而Lanczos推送是一种时间复杂度与图规模无关的局部算法。具体而言，我们证明Lanczos迭代的时间复杂度为$\tilde{O}(\sqrtκ m)$（$m$为图的边数，$\tilde{O}$表示忽略对数项的复杂度），与先前基于幂迭代的全局方法相比实现了$\sqrtκ$的加速。对于Lanczos推送，我们证明在某些温和且常见的假设下，其时间复杂度为$\tilde{O}(κ^{2.75})$，这相较于最先进的基于随机游走的局部算法实现了$κ^{0.25}$的显著提升。我们在八个具有不同规模和统计特性的真实数据集上进行了广泛实验，验证了所提算法的有效性，结果表明Lanczos迭代和Lanczos推送在效率和精度方面均显著优于现有最先进方法。