A Survey on Hypergraph Mining: Patterns, Tools, and Generators

TL;DR

This survey addresses mining structural patterns in real-world hypergraphs, which model higher-order group interactions using hyperedges of variable size. It develops a unified taxonomy of static and dynamic patterns across node-, hyperedge-, subhypergraph-, and hypergraph-levels, grounding these patterns in concrete measures such as node degree $d(v;H)$ and hyperedge size $|e|$. It reviews mining tools (null models, structural elements, and structural quantities) and catalogs generators that reproduce observed patterns, distinguishing full-hypergraph and sub-hypergraph, as well as static and dynamic models. The framework supports advances in algorithm design, machine learning on hypergraphs, and analysis of generalized hypergraphs, with practical implications for benchmarking, simulation, and understanding higher-order systems in diverse domains.

Abstract

Hypergraphs, which belong to the family of higher-order networks, are a natural and powerful choice for modeling group interactions in the real world. For example, when modeling collaboration networks, which may involve not just two but three or more people, the use of hypergraphs allows us to explore beyond pairwise (dyadic) patterns and capture groupwise (polyadic) patterns. The mathematical complexity of hypergraphs offers both opportunities and challenges for hypergraph mining. The goal of hypergraph mining is to find structural properties recurring in real-world hypergraphs across different domains, which we call patterns. To find patterns, we need tools. We divide hypergraph mining tools into three categories: (1) null models (which help test the significance of observed patterns), (2) structural elements (i.e., substructures in a hypergraph such as open and closed triangles), and (3) structural quantities (i.e., numerical tools for computing hypergraph patterns such as transitivity). There are also hypergraph generators, whose objective is to produce synthetic hypergraphs that are a faithful representation of real-world hypergraphs. In this survey, we provide a comprehensive overview of the current landscape of hypergraph mining, covering patterns, tools, and generators. We provide comprehensive taxonomies for each and offer in-depth discussions for future research on hypergraph mining.

Figures (10)

• Figure 1: Group interactions are naturally modeled as hypergraphs. Here, the co-authorship among seven authors across four publications is modeled as a hypergraph with seven nodes and four hyperedges.
• Figure 2: The overall structure of this survey.
• Figure 3: Dyadic projections (B\ref{['background:dyadic_projections']}) are applied to a hypergraph to obtain its clique expansion and star expansion.
• Figure 4: A temporal hypergraph (B\ref{['background:temporal_hypergraphs']}) evolves over time as new hyperedges (colored red) are added. Note that the same hyperedge can be added multiple times at different time stamps. Here, at $t_4 = 12$ and $t_5 = 14$, the same hyperedge $\{6,7,8\}$ is added.
• Figure 5: A taxonomy for tools used for defining and mining structural patterns (Section \ref{['sec:tools']}).

Theorems & Definitions (4)

• definition 1: (Induced) subhypergraphs
• definition 2: Clique expansions
• definition 3: Star expansions
• definition 4: Temporal hypergraphs