We study the problem of efficiently approximating the \textit{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 $\epsilon$. The previous best algorithm for this problem has a worst-case computational complexity of $\tilde{O}\left(\frac{L_{\max}^3}{\epsilon^2 d^2}\right)$, where the value of $L_{\max}$ depends on the mixing time of random walks on $G$, $d = \min\{d(s), d(t)\}$, and $d(s)$, $d(t)$ denote the degrees of nodes $s$ and $t$, respectively. We improve this complexity to $\tilde{O}\left(\min\left\{\frac{L_{\max}^{7/3}}{\epsilon^{2/3}}, \frac{L_{\max}^3}{\epsilon^2d^2}, mL_{\max}\right\}\right)$, achieving a theoretical improvement of $\tilde{O}\left(\max\left\{\frac{L_{\max}^{2/3}}{\epsilon^{4/3} d^2}, 1, \frac{L_{\max}^2}{\epsilon^2 d^2 m}\right\}\right)$ over previous results. Here, $m$ denotes the number of edges. Given that $L_{\max}$ is often very large in real-world networks (e.g., $L_{\max} > 10^4$), our improvement on $L_{\max}$ 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 $10\times$ to $1000\times$ speedup in running time while maintaining the same absolute error compared to baseline methods.

翻译：我们研究了在无向图上高效近似计算\textit{有效电阻}（ER）的问题，其中ER是一种广泛使用的节点邻近度度量，在图谱稀疏化、多类图聚类、网络鲁棒性分析、图机器学习等领域有重要应用。具体而言，给定无向图$G$中的任意节点$s$和$t$，我们的目标是高效估计节点$s$与$t$之间的ER值$R(s,t)$，并确保较小的绝对误差$\epsilon$。先前解决该问题的最佳算法在最坏情况下的计算复杂度为$\tilde{O}\left(\frac{L_{\max}^3}{\epsilon^2 d^2}\right)$，其中$L_{\max}$的值取决于图$G$上随机游走的混合时间，$d = \min\{d(s), d(t)\}$，而$d(s)$、$d(t)$分别表示节点$s$和$t$的度数。我们将该复杂度改进为$\tilde{O}\left(\min\left\{\frac{L_{\max}^{7/3}}{\epsilon^{2/3}}, \frac{L_{\max}^3}{\epsilon^2d^2}, mL_{\max}\right\}\right)$，相较于先前结果实现了$\tilde{O}\left(\max\left\{\frac{L_{\max}^{2/3}}{\epsilon^{4/3} d^2}, 1, \frac{L_{\max}^2}{\epsilon^2 d^2 m}\right\}\right)$的理论改进。此处，$m$表示边的数量。考虑到在实际网络中$L_{\max}$通常非常大（例如$L_{\max} > 10^4$），我们对$L_{\max}$的改进是显著的，尤其对于现实世界网络。我们还在真实世界和合成的图数据集上进行了大量实验，以实证证明我们方法的优越性。实验结果表明，在保持相同绝对误差的前提下，与基线方法相比，我们的方法在运行时间上实现了$10$倍到$1000$倍的加速。