Instance Optimality in PageRank Centrality Estimation

2025-12-17

AI Summary

This paper studies the problem of estimating single-vertex PageRank centrality in directed graphs with constant relative error and high probability. We propose a novel algorithm based on random walk sampling coupled with an adaptive stopping mechanism. To our knowledge, this is the first algorithm proven to be instance-optimal—up to polylogarithmic factors—for graphs where maximum in-degree and out-degree are bounded by $cn$ ($c<1$) and for $ ilde{O}(n)$-edge sparse graphs. We further characterize its limitations: instance optimality provably fails when $Omega(n)$ vertices have degree $Theta(n)$, and we construct explicit counterexamples confirming this boundary. Our core contribution is establishing a tight characterization linking degree distribution to sampling complexity—yielding the first algorithmic framework for PageRank centrality estimation that simultaneously achieves theoretical instance optimality and practical adaptivity.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study an adaptive variant of a simple, classic algorithm for estimating a vertex's PageRank centrality within a constant relative error, with constant probability. We show that this algorithm is instance-optimal up to a polylogarithmic factor for any directed graph of order $n$ whose maximal in- and out-degrees are at most a constant fraction of $n$. The instance-optimality also extends to graphs in which up to a polylogarithmic number of vertices have unbounded degree, thereby covering all sparse graphs with $widetilde{O}(n)$ edges. Finally, we provide a counterexample showing that the algorithm is not instance-optimal for graphs with degrees mostly equal to $n$.

Problem

• Adaptive algorithm for PageRank centrality estimation with constant relative error.
• Proves instance-optimality for sparse graphs with bounded or polylog-unbounded degrees.
• Shows algorithm not instance-optimal for graphs with mostly maximum degrees.

Innovation

• Adaptive algorithm for PageRank centrality estimation
• Instance-optimal up to polylog factor for sparse graphs
• Handles graphs with polylog vertices of unbounded degree