A note on approximating the average degree of bounded arboricity graphs

2026-03-09

AI Summary

This work addresses the problem of efficiently estimating the average degree of graphs with bounded arboricity in sublinear time. Under the standard graph access model—supporting vertex sampling, degree queries, and neighbor queries—we present a simple (1+ε)-approximation algorithm with query complexity O(ε⁻²α/d), where α is an upper bound on the graph’s arboricity and d denotes the average degree. By explicitly uncovering the implicit algorithmic structure underlying prior approaches, our method eliminates the logarithmic overhead incurred by parameter search, thereby enhancing both theoretical clarity and efficiency. Furthermore, we refine the algorithm to achieve a query complexity of O(ε⁻²√(n/d)), maintaining the same approximation guarantee while improving adaptability to sparse graphs.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Heuristic Search

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• 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

Abstract

Estimating the average degree of graph is a classic problem in sublinear graph algorithm. Eden, Ron, and Seshadhri (ICALP 2017, SIDMA 2019) gave a simple algorithm for this problem whose running time depended on the graph arboricity, but the underlying simplicity and associated analysis were buried inside the main result. Moreover, the description there loses logarithmic factors because of parameter search. The aim of this note is to give a full presentation of this algorithm, without these losses. Consider standard access (vertex samples, degree queries, and neighbor queries) to a graph $G = (V,E)$ of arboricity at most $\alpha$. Let $d$ denote the average degree of $G$. We describe an algorithm that gives a $(1+\varepsilon)$-approximation to $d$ degree using $O(\varepsilon^{-2}\alpha/d)$ queries. For completeness, we modify the algorithm to get a $O(\varepsilon^{-2} \sqrt{n/d})$ query

Problem

• average degree
• bounded arboricity
• sublinear algorithms
• graph approximation

Innovation

• sublinear algorithms
• average degree estimation
• bounded arboricity
• graph queries
• approximation algorithms