Personalized PageRank Estimation in Undirected Graphs

2026-02-11

AI Summary

This study addresses the problem of efficiently estimating personalized PageRank (PPR) under a threshold condition in undirected graphs, covering three variants: single-source, single-target, and single-pair queries. Leveraging the symmetric structure of undirected graphs and combining random walk theory with graph access model analysis, the authors propose a novel algorithm and establish matching lower bounds on computational complexity. For a given relative error and failure probability, they achieve— for the first time—theoretical guarantees that are tight up to logarithmic factors for all three query variants, both in worst-case and average-case settings. This work thus fully characterizes the fundamental limits of PPR estimation in undirected graphs.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Planning, Routing, and Scheduling: Planning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Given an undirected graph $G=(V, E)$, the Personalized PageRank (PPR) of $t\in V$ with respect to $s\in V$, denoted $\pi(s,t)$, is the probability that an $\alpha$-discounted random walk starting at $s$ terminates at $t$. We study the time complexity of estimating $\pi(s,t)$ with constant relative error and constant failure probability, whenever $\pi(s,t)$ is above a given threshold parameter $\delta\in(0,1)$. We consider common graph-access models and furthermore study the single source, single target, and single node (PageRank centrality) variants of the problem. We provide a complete characterization of PPR estimation in undirected graphs by giving tight bounds (up to logarithmic factors) for all problems and model variants in both the worst-case and average-case setting. This includes both new upper and lower bounds. Tight bounds were recently obtained by Bertram, Jensen, Thorup, Wang, and Yan for directed graphs. However, their lower bound constructions rely on asymmetry and therefore do not carry over to undirected graphs. At the same time, undirected graphs exhibit additional structure that can be exploited algorithmically. Our results resolve the undirected case by developing new techniques that capture both aspects, yielding tight bounds.

Problem

• Personalized PageRank
• undirected graphs
• time complexity
• random walks
• estimation

Innovation

• Personalized PageRank
• undirected graphs
• tight bounds
• random walks
• graph algorithms