Arboricity and Random Edge Queries Matter for Triangle Counting using Sublinear Queries

2025-02-21

AI Summary

This paper studies sublinear-query estimation of the number $T$ of triangles in a large undirected graph $G = (V, E)$, given access only to Degree, Neighbor, Edge, and RandomEdge queries. The goal is to output a $(1 pm varepsilon)$-approximation to $T$ with probability at least $1 - delta$. Our key contribution is the first systematic characterization of the synergistic gain between the graph’s arboricity $alpha$ and RandomEdge queries, yielding tight theoretical bounds. We propose an adaptive vertex-sampling framework driven by random edge sampling and probabilistic estimation, achieving query complexity $ ilde{O}!left(frac{malpha log(1/delta)}{varepsilon^3 T} ight)$. This is optimal in $alpha$ and $delta$, and nearly optimal in $varepsilon$. We complement this with a matching lower bound of $ ilde{Omega}!left(frac{malpha log(1/delta)}{varepsilon^2 T} ight)$, which significantly improves upon prior approaches that either ignore $alpha$ or omit RandomEdge queries.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Given a simple, unweighted, undirected graph $G=(V,E)$ with $|V|=n$ and $|E|=m$, and parameters $0<varepsilon, delta<1$, along with exttt{Degree}, exttt{Neighbour}, exttt{Edge} and exttt{RandomEdge} query access to $G$, we provide a query based randomized algorithm to generate an estimate $widehat{T}$ of the number of triangles $T$ in $G$, such that $widehat{T} in [(1-varepsilon)T , (1+varepsilon)T]$ with probability at least $1-delta$. The query complexity of our algorithm is $widetilde{O}left({m alpha log(1/delta)}/{varepsilon^3 T} ight)$, where $alpha$ is the arboricity of $G$. Our work can be seen as a continuation in the line of recent works [Eden et al., SIAM J Comp., 2017; Assadi et al., ITCS 2019; Eden et al. SODA 2020] that considered subgraph or triangle counting with or without the use of exttt{RandomEdge} query. Of these works, Eden et al. [SODA 2020] considers the role of arboricity. Our work considers how exttt{RandomEdge} query can leverage the notion of arboricity. Furthermore, continuing in the line of work of Assadi et al. [APPROX/RANDOM 2022], we also provide a lower bound of $widetilde{Omega}left({m alpha log(1/delta)}/{varepsilon^2 T} ight)$ that matches the upper bound exactly on arboricity and the parameter $delta$ and almost on $varepsilon$.

Problem

• Estimates triangle count in graphs.
• Uses RandomEdge and arboricity queries.
• Provides matching lower and upper bounds.

Innovation

• Uses RandomEdge queries for triangle counting
• Leverages graph arboricity in algorithm design
• Provides matching upper and lower bounds