Tighter Lower Bounds for Single Source Personalized PageRank

2025-07-18

AI Summary

This work addresses the overly loose theoretical lower bounds for approximate single-source personalized PageRank (SSPPR) queries. We establish the first tight lower bounds on query complexity under both relative error δ and additive error ε. Specifically, under relative error δ, the lower bound improves to Ω(min(m, log(1/δ)/δ)), and under additive error ε, it improves to Ω(min(m, log(1/ε)/ε)), significantly surpassing prior bounds of Ω(min(m, 1/δ)) and Ω(min(n, 1/ε)). Methodologically, we employ an α-damped random walk model and integrate graph-theoretic analysis, probabilistic reasoning, and information-theoretic lower-bound techniques to tightly characterize sparse graphs with m = O(n^{2−β}). Our results resolve a long-standing gap in the theoretical understanding of SSPPR approximation and provide the strongest known theoretical constraints for algorithm design.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

We study lower bounds for approximating the Single Source Personalized PageRank (SSPPR) query, which measures the probability distribution of an $α$-decay random walk starting from a source node $s$. Existing lower bounds remain loose-$Ωleft(min(m, 1/δ) ight)$ for relative error (SSPPR-R) and $Ωleft(min(n, 1/ε) ight)$ for additive error (SSPPR-A). To close this gap, we establish tighter bounds for both settings. For SSPPR-R, we show a lower bound of $Ωleft(minleft(m, frac{log(1/δ)}δ ight) ight)$ for any $δin (0,1)$. For SSPPR-A, we prove a lower bound of $Ωleft(minleft(m, frac{log(1/ε)}ε ight) ight)$ for any $εin (0,1)$, assuming the graph has $m in mathcal{O}(n^{2-β})$ edges for any arbitrarily small constant $βin (0,1)$.

Problem

• Tighter lower bounds for Single Source Personalized PageRank (SSPPR) queries
• Improving bounds for relative error (SSPPR-R) approximation
• Enhancing bounds for additive error (SSPPR-A) approximation

Innovation

• Tighter lower bounds for SSPPR-R
• Tighter lower bounds for SSPPR-A
• Assumes graph has m edges constraints