Mixing Time Matters: Accelerating Effective Resistance Estimation via Bidirectional Method
Guanyu Cui, Hanzhi Wang, Zhewei Wei
TL;DR
This paper tackles the problem of efficiently estimating the Single-Pair Effective Resistance $R(s,t)$ in undirected graphs, a task central to graph sparsification, clustering, and robust network analysis. It introduces BiSPER, a bidirectional estimator that combines a redesigned forward push with adaptive Monte Carlo sampling, leveraging $L_{\max}$-truncation and a symmetry property of multi-step transition probabilities to form an unbiased estimator for $R_{L_{\max}}(s,t)$. The authors prove correctness and derive a near-optimal time bound $\tilde{O}(\min\{L_{\max}^{7/3}/\epsilon^{2/3}, L_{\max}^3/(\epsilon^2 d^2), mL_{\max}\})$, and demonstrate up to $10\times$ to $1000\times$ empirical speedups over prior methods while preserving absolute error guarantees. Extensive experiments on real-world SNAP graphs and synthetic networks show substantial improvements in efficiency and accuracy, particularly when the mixing time is high (large $L_{\max}$), highlighting BiSPER’s practical impact for large-scale graph analysis and downstream applications in graph learning and robustness screening.
Abstract
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 $ε$. The previous best algorithm for this problem has a worst-case computational complexity of $\tilde{O}\left(\frac{L_{\max}^3}{ε^2 d^2}\right)$, 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 $\tilde{O}\left(\min\left\{\frac{L_{\max}^{7/3}}{ε^{2/3}}, \frac{L_{\max}^3}{ε^2d^2}, mL_{\max}\right\}\right)$, achieving a theoretical improvement of $\tilde{O}\left(\max\left\{\frac{L_{\max}^{2/3}}{ε^{4/3} d^2}, 1, \frac{L_{\max}^2}{ε^2 d^2 m}\right\}\right)$ 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.