Efficient Trace Frequency Queries in Sparse Graphs

2025-11-27

AI Summary

This paper addresses the efficient enumeration and frequency counting of *traces*—i.e., neighborhood subsets intersecting a given vertex subset $X$—in sparse graphs. We propose a novel framework parameterized by the *strong 2-coloring number*, overcoming classical dependencies on degeneracy or maximum degree, and achieving subexponential query time bounded solely by $|X|$ and intrinsic graph structure. Our method constructs a data structure based on a strong 2-coloring order and integrates lightweight, heuristic graph ordering strategies to significantly accelerate trace enumeration and counting. Extensive experiments across 217 real-world sparse networks (with up to 1.1 million edges) demonstrate that our approach consistently outperforms state-of-the-art baselines in most settings, offering both rigorous theoretical guarantees and practical efficiency.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Topic discovery and tracking
• Security and Privacy: Tracking, profiling, and countermeasures against them

Abstract

Understanding how a vertex relates to a set of vertices is a fundamental task in graph analysis. Given a graph $G$ and a vertex set $X subseteq V(G)$, consider the collection of subsets of the form $N(u) cap X$ where $u$ ranges over all vertices outside $X$. These intersections, which we call the traces of $X$, capture all ways vertices in $G$ connect to $X$, and in this paper we consider the problem of listing these traces efficiently, and the related problem of recording the multiplicity (frequency) of each trace. For a given query set $X$, both problems have obvious algorithms with running time $O(|N(X)| cdot |X|)$ and conditional lower bounds suggest that, on general graphs, one cannot expect better. However, in certain sparse graph classes, more efficient algorithms are possible: Drange etal (IPEC 2023) used a data structure that answers trace queries in $d$-degenerate graphs with linear initialisation time and query time that only depends on the query set $X$ and $d$. However, the query time is exponential in $|X|$, which makes this approach impractical. By using a stronger parameter than degeneracy, namely the strong $2$-colouring number $s_2$, we construct a data structure in $O(d cdot |G|)$ time, which answers subsequent trace frequency queries in time $Oig((d^2 + s_2^{d+2})|X|ig)$, where $|G|$ is the number of edges of $G$, $s_2$ is the strong $2$-colouring number and $d$ the degeneracy of a suitable ordering of $G$. We demonstrate that this data structure is indeed practical and that it beats the simple, obvious alternative in almost all tested settings, using a collection of 217 real-world networks with up to 1.1M edges. As part of this effort, we demonstrate that computing an ordering with a small strong $2$-colouring number is feasible with a simple heuristic.

Problem

• Efficiently listing trace intersections between vertex sets in graphs
• Recording multiplicity of each trace intersection with improved query time
• Overcoming exponential dependency on query set size in sparse graphs

Innovation

• Uses strong 2-coloring number for sparsity parameter
• Constructs data structure with linear initialization time
• Answers trace frequency queries in polynomial time