Counting hypertriangles through hypergraph orientations

2026-01-07

arXiv.org

AI Summary

This work addresses the need for efficient higher-order structural analysis in complex networks by studying the counting of hypertriangles—patterns formed by three pairwise-intersecting hyperedges—in hypergraphs. Inspired by graph orientation and degeneracy-based algorithms, the authors generalize the concepts of graph orientation and degeneracy ordering to hypergraphs for the first time, proposing DITCH, a provably efficient counting algorithm. DITCH integrates hypergraph orientation, degeneracy ordering, and combinatorial enumeration to effectively handle the diverse intersection structures inherent in hypertriangles. Experimental results demonstrate that DITCH achieves speedups of 10–100× over state-of-the-art methods while substantially reducing memory consumption.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• 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

Counting the number of small patterns is a central task in network analysis. While this problem is well studied for graphs, many real-world datasets are naturally modeled as hypergraphs, motivating the need for efficient hypergraph motif counting algorithms. In particular, we study the problem of counting hypertriangles - collections of three pairwise-intersecting hyperedges. These hypergraph patterns have a rich structure with multiple distinct intersection patterns unlike graph triangles. Inspired by classical graph algorithms based on orientations and degeneracy, we develop a theoretical framework that generalizes these concepts to hypergraphs and yields provable algorithms for hypertriangle counting. We implement these ideas in DITCH (Degeneracy Inspired Triangle Counter for Hypergraphs) and show experimentally that it is 10-100x faster and more memory efficient than existing state-of-the-art methods.

Problem

• hypergraph
• hypertriangle counting
• network analysis
• motif counting

Innovation

• hypergraph
• hypertriangle counting
• orientation
• degeneracy
• motif enumeration