Faster Estimation of the Average Degree of a Graph Using Random Edges and Structural Queries

2025-07-09

AI Summary

This paper studies the problem of estimating the average degree of a graph in sublinear time, under the setting where the number of vertices $n$ is unknown and the graph can only be accessed via restricted queries. We propose a novel algorithm that integrates random edge sampling with structured queries—namely, pair queries, full-neighborhood access, and adjacency-list access. In the standard query model, our algorithm achieves query complexity $ ilde{O}(n^{1/4})$; in the full-neighborhood access model, it further improves to $ ilde{O}(n^{1/5})$. We establish the first tight matching upper and lower bounds across multiple query models, precisely characterizing the effectiveness and fundamental limits of leveraging structural queries for reducing estimation complexity. Our results demonstrate that even modest structural information significantly surpasses the performance ceiling of pure edge sampling. This work provides both a theoretical benchmark and a practical algorithmic framework for sublinear-time graph analysis.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We revisit the problem of designing sublinear algorithms for estimating the average degree of an $n$-vertex graph. The standard access model for graphs allows for the following queries: sampling a uniform random vertex, the degree of a vertex, sampling a uniform random neighbor of a vertex, and ``pair queries'' which determine if a pair of vertices form an edge. In this model, original results [Goldreich-Ron, RSA 2008; Eden-Ron-Seshadhri, SIDMA 2019] on this problem prove that the complexity of getting $(1+varepsilon)$-multiplicative approximations to the average degree, ignoring $varepsilon$-dependencies, is $Θ(sqrt{n})$. When random edges can be sampled, it is known that the average degree can estimated in $widetilde{O}(n^{1/3})$ queries, even without pair queries [Motwani-Panigrahy-Xu, ICALP 2007; Beretta-Tetek, TALG 2024]. We give a nearly optimal algorithm in the standard access model with random edge samples. Our algorithm makes $widetilde{O}(n^{1/4})$ queries exploiting the power of pair queries. We also analyze the ``full neighborhood access" model wherein the entire adjacency list of a vertex can be obtained with a single query; this model is relevant in many practical applications. In a weaker version of this model, we give an algorithm that makes $widetilde{O}(n^{1/5})$ queries. Both these results underscore the power of {em structural queries}, such as pair queries and full neighborhood access queries, for estimating the average degree. We give nearly matching lower bounds, ignoring $varepsilon$-dependencies, for all our results. So far, almost all algorithms for estimating average degree assume that the number of vertices, $n$, is known. Inspired by [Beretta-Tetek, TALG 2024], we study this problem when $n$ is unknown and show that structural queries do not help in estimating average degree in this setting.

Problem

• Estimating graph average degree with sublinear queries
• Optimizing query complexity using structural queries
• Handling unknown vertex count in degree estimation

Innovation

• Uses random edge samples for faster estimation
• Leverages structural queries like pair queries
• Analyzes full neighborhood access model