A Simple Active-Set Method for PageRank-Based Local Graph Clustering

2026-08-17

AI Summary

This study addresses the limitations of approximate PageRank computation in local graph clustering, where performance is typically constrained by graph size and parameter dependencies. We propose an active-set-based approximation algorithm that integrates active set expansion, push operations, and a near-linear time SDD solver to decouple runtime from graph size while optimizing dependence on regularization parameters. Theoretical analysis demonstrates that the algorithm achieves approximation in Õ(1/ε²) time with high probability, establishing a novel conductance-volume runtime trade-off. These contributions significantly enhance both computational efficiency and theoretical guarantees for local clustering on large-scale graphs, overcoming fundamental bottlenecks in existing PageRank-based approaches.

Abstract

Local graph clustering aims to find a well-connected cluster near a given seed node without exploring the entire graph. A key step in the classic local clustering algorithm of Andersen, Chung, and Lang (ACL; Internet Math. 2007) is to approximate the PageRank vector from the seed node. Their local push method computes an ACL $\varepsilon$-approximate PageRank vector with teleportation parameter $α$ in $O\bigl(1/(α\varepsilon)\bigr)$ time. We give an algorithm that computes an ACL $\varepsilon$-approximate PageRank vector in $\widetilde{O}\bigl(1 / \varepsilon^2\bigr)$ time with high probability. This bound is independent of the graph size and has only a polylogarithmic dependence on $1 / α$, albeit with a quadratic dependence on $1 / \varepsilon$. As a direct consequence, we obtain a new running-time tradeoff between the target conductance and target volume in local graph clustering. Our method also applies to the optimization problem of $\ell_1$-regularized PageRank and computes an additive approximate minimizer with a polylogarithmic dependence on $1/α$, improving the $1/\sqrtα$ dependence in the previous bound of Martínez-Rubio, Wirth, and Pokutta (COLT 2023). Our algorithm is based on an intuitive process that maintains a growing active set of nodes: it performs push operations on the current set until convergence and then expands the set and repeats the process if necessary. We show that for each active set, the corresponding limiting state is the solution to a symmetric diagonally dominant (SDD) linear system on the set. We apply nearly-linear-time SDD solvers to these systems and prove that the approximation preserves the properties of the push process.

Problem

• Local Graph Clustering
• Approximate PageRank
• Active-Set Method
• l1-regularized PageRank

Innovation

• Local Graph Clustering
• PageRank
• Active-Set Method
• SDD Linear System
• Approximation Algorithm