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Efficient Computation of Hyper-triangles on Hypergraphs

Haozhe Yin, Kai Wang, Wenjie Zhang, Ying Zhang, Ruijia Wu, Xuemin Lin

TL;DR

This work tackles the computation of hyper-triangles in hypergraphs, introducing a two-step, hyperwedge-based framework that transforms the problem into wedge classification followed by pattern-specific triangle assembly. It provides exact algorithms with specialized treatments for the TTT class and for the other pattern classes, along with complexity and parallelization analyses. To handle large-scale data, it also offers unbiased approximate counting schemes, including a targeted Advanced algorithm that improves accuracy for specific pattern classes. Additionally, the authors propose a fine-grained hypergraph clustering coefficient to reflect pattern-dependent connectivity and validate the approach on 11 real-world datasets, showing substantial speedups and reliable estimates. The combination of exact, approximate, and pattern-aware metrics enables efficient, scalable analysis of higher-order interactions in complex networks.

Abstract

Hypergraphs, which use hyperedges to capture groupwise interactions among different entities, have gained increasing attention recently for their versatility in effectively modeling real-world networks. In this paper, we study the problem of computing hyper-triangles (formed by three fully-connected hyperedges), which is a basic structural unit in hypergraphs. Although existing approaches can be adopted to compute hyper-triangles by exhaustively examining hyperedge combinations, they overlook the structural characteristics distinguishing different hyper-triangle patterns. Consequently, these approaches lack specificity in computing particular hyper-triangle patterns and exhibit low efficiency. In this paper, we unveil a new formation pathway for hyper-triangles, transitioning from hyperedges to hyperwedges before assembling into hyper-triangles, and classify hyper-triangle patterns based on hyperwedges. Leveraging this insight, we introduce a two-step framework to reduce the redundant checking of hyperedge combinations. Under this framework, we propose efficient algorithms for computing a specific pattern of hyper-triangles. Approximate algorithms are also devised to support estimated counting scenarios. Furthermore, we introduce a fine-grained hypergraph clustering coefficient measurement that can reflect diverse properties of hypergraphs based on different hyper-triangle patterns. Extensive experimental evaluations conducted on 11 real-world datasets validate the effectiveness and efficiency of our proposed techniques.

Efficient Computation of Hyper-triangles on Hypergraphs

TL;DR

This work tackles the computation of hyper-triangles in hypergraphs, introducing a two-step, hyperwedge-based framework that transforms the problem into wedge classification followed by pattern-specific triangle assembly. It provides exact algorithms with specialized treatments for the TTT class and for the other pattern classes, along with complexity and parallelization analyses. To handle large-scale data, it also offers unbiased approximate counting schemes, including a targeted Advanced algorithm that improves accuracy for specific pattern classes. Additionally, the authors propose a fine-grained hypergraph clustering coefficient to reflect pattern-dependent connectivity and validate the approach on 11 real-world datasets, showing substantial speedups and reliable estimates. The combination of exact, approximate, and pattern-aware metrics enables efficient, scalable analysis of higher-order interactions in complex networks.

Abstract

Hypergraphs, which use hyperedges to capture groupwise interactions among different entities, have gained increasing attention recently for their versatility in effectively modeling real-world networks. In this paper, we study the problem of computing hyper-triangles (formed by three fully-connected hyperedges), which is a basic structural unit in hypergraphs. Although existing approaches can be adopted to compute hyper-triangles by exhaustively examining hyperedge combinations, they overlook the structural characteristics distinguishing different hyper-triangle patterns. Consequently, these approaches lack specificity in computing particular hyper-triangle patterns and exhibit low efficiency. In this paper, we unveil a new formation pathway for hyper-triangles, transitioning from hyperedges to hyperwedges before assembling into hyper-triangles, and classify hyper-triangle patterns based on hyperwedges. Leveraging this insight, we introduce a two-step framework to reduce the redundant checking of hyperedge combinations. Under this framework, we propose efficient algorithms for computing a specific pattern of hyper-triangles. Approximate algorithms are also devised to support estimated counting scenarios. Furthermore, we introduce a fine-grained hypergraph clustering coefficient measurement that can reflect diverse properties of hypergraphs based on different hyper-triangle patterns. Extensive experimental evaluations conducted on 11 real-world datasets validate the effectiveness and efficiency of our proposed techniques.

Paper Structure

This paper contains 19 sections, 5 theorems, 1 equation, 10 figures, 3 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Problem definition
  3. Exact Algorithms
  4. The Two-step Framework
  5. Algorithms for TTT class
  6. Algorithms for other classes
  7. Complexity Analysis
  8. Parallelization
  9. Approximate Counting
  10. The Basic Algorithm
  11. The Advanced Algorithm
  12. The Fine-grained Hypergraph Clustering Coefficient
  13. Experiments
  14. Experimental Settings
  15. Case Study
  16. ...and 4 more sections

Key Result

Lemma 3.1

For two hyperwedges $\hbox{o}rigin=c]{270}{$<$}^t_{ij}, \hbox{o}rigin=c]{270}{$<$}^t_{ik}\in \hbox{o}rigin=c]{270}{$<$}^t$ where $\mathcal{O}(\hbox{o}rigin=c]{270}{$<$}^t_{ij})<\mathcal{O}(\hbox{o}rigin=c]{270}{$<$}^t_{ik})$. if the set of common vertices between $e_i$ and $e_j$ equals to the set of

Figures (10)

  • Figure 1: Graph model comparison.
  • Figure 2: All the patterns for hyper-triangles.
  • Figure 3: A sparse hypergraph
  • Figure 4: Clustering coefficient comparison
  • Figure 5: Real data for co-authorship relations
  • ...and 5 more figures

Theorems & Definitions (10)

  • Definition 2.1: Hypergraph
  • Definition 2.2: Hyperwedge
  • Definition 2.3: Intersection/Inclusion hyperwedge
  • Definition 2.4: Hyper-triangle
  • Lemma 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Definition 5.1