Mixing Time Matters: Accelerating Effective Resistance Estimation via Bidirectional Method

2025-03-04

AI Summary

This paper addresses the problem of efficiently approximating effective resistance (ER) between node pairs in undirected graphs, aiming to estimate ( R(s,t) ) for any source–target pair ( s ext{--}t ) within absolute error ( varepsilon ), supporting spectral sparsification, multi-way clustering, and graph learning. We propose the first algorithm that reduces the dependence on the graph’s maximum mixing time ( L_{max} ) from ( O(L_{max}^3) ) to ( widetilde{O}(L_{max}^{7/3}/varepsilon^{2/3}) ), achieving a theoretical speedup of ( widetilde{O}ig(max{L_{max}^{2/3}/(varepsilon^{4/3}d^2),, L_{max}^2/(varepsilon^2 d^2 m)}ig) ). Our method integrates bidirectional random walk sampling, variance reduction, and adaptive truncation—overcoming the high-order dependency bottleneck inherent in conventional unidirectional approaches. Empirically, it delivers 10×–1000× speedups on real-world and synthetic graphs while preserving estimation accuracy. The worst-case complexity is improved to ( widetilde{O}ig(min{L_{max}^{7/3}/varepsilon^{2/3},, L_{max}^3/(varepsilon^2 d^2),, m L_{max}}ig) ).

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

We study the problem of efficiently approximating the extit{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 $ ilde{O}left(frac{L_{max}^3}{epsilon^2 d^2} ight)$, 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 $ ilde{O}left(minleft{frac{L_{max}^{7/3}}{epsilon^{2/3}}, frac{L_{max}^3}{epsilon^2d^2}, mL_{max} ight} ight)$, achieving a theoretical improvement of $ ilde{O}left(maxleft{frac{L_{max}^{2/3}}{epsilon^{4/3} d^2}, 1, frac{L_{max}^2}{epsilon^2 d^2 m} ight} ight)$ 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 $10 imes$ to $1000 imes$ speedup in running time while maintaining the same absolute error compared to baseline methods.

Problem

• Efficiently estimate effective resistance in undirected graphs.
• Improve computational complexity for large real-world networks.
• Achieve significant speedup while maintaining accuracy.

Innovation

• Bidirectional method accelerates effective resistance estimation.
• Improved computational complexity via optimized mixing time utilization.
• Achieves significant speedup in real-world network applications.