Counting hypertriangles through hypergraph orientations

TL;DR

This work addresses exact counting of hypergraph motifs, focusing on hypertriangles, and introduces a degeneracy-oriented framework that generalizes graph orientations to hypergraphs. By defining directed acyclic hypergraphs and a hyperedge degeneracy κ(G), the authors develop algorithms whose runtimes scale with sums of squared out-degrees, specifically O(∑v d_v(d^+_v)^2), enabling efficient, exact counting. The DITCH implementation leverages triangle-, containment-, and star-based counting, plus open-pattern handling, and achieves 10–100x speedups with far lower memory usage than prior methods on real datasets, including large-scale Tg-ubuntu. These results demonstrate the practicality of orientation-based techniques for hypergraph motif counting and point to future extensions to larger motifs and more nuanced intersection conditions.

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.

Figures (9)

• Figure 1: Runtime and memory comparisons: we compare our algorithm DITCH with the existing best MoCHy-ELeJiSh20 and state-of-the-art Exact-advYiWaZh+24 algorithms. We test on a collection of ten datasets, and set a memory limit of 9GB. Any bar that is not present means that the algorithm ran out of memory. Across all datasets, DITCH is 10-100x faster and more memory efficient than previous methods.
• Figure 2: Hypertriangles, as defined by LeJiSh20. The three circles represent the distinct hyperedges, and regions are colored if they are non-empty. Regions in exactly one set are colored green, two sets are colored blue, and three sets are colored red. The 20 closed patterns are technically the hypertriangles, whereas the 6 open patterns are akin to paths.
• Figure 3: Degree and Outdegree Distributions: For four datasets, we plot the difference between the standard degree distribution and outdegree distribution of the oriented hypergraph. Each point on the $x$-axis is a degree, and the $y$-axis is the frequency of that degree, plotting in log-log. Observe how the outdegree frequencies are significantly smaller for higher degrees, which cuts down the search space for hypertriangles. For the co-author datasets marked "A", the reduction of the tail is dramatic.
• Figure 4: Examples of hypergraph patterns that are found using the Triangle based algorithm.
• Figure 5: The two possible cases in a star-based hypertriangle.

Theorems & Definitions (39)

• definition 1: Hyperedge degeneracy
• theorem 1
• lemma 1
• proof
• lemma 2
• proof
• lemma 3
• proof
• lemma 4
• proof