A survey of simplicial, relative, and chain complex homology theories for hypergraphs

TL;DR

This survey consolidates nine nontrivial ways to derive topological objects from hypergraphs—primarily simplicial and chain- complex constructions—and studies their associated homology theories. It systematically defines each construction (closure, restricted/relative barycentric subdivision, polar complex, embedded, path, weighted nerve, and categorification-based chromatic and magnitude homologies), proving basic functoriality where possible and exploring properties like duality and recoverability. The authors provide intuition through running examples, discuss computational considerations, and highlight how hypergraph structure (blocks, edges, and subedges) shapes homological invariants. The work emphasizes open questions, potential persistence frameworks, and practical relevance for applications in data science, biology, and networks, inviting further development of software tools and theoretical refinements.

Abstract

Hypergraphs have seen widespread application in network and data science communities in recent years. We present a survey of recent work to construct auxiliary structures from hypergraphs -- specifically simplicial, relative, and chain complexes -- that can be used to build homology theories for hypergraphs. We define and describe nine different constructions and their associated homology theories. We discuss some interesting properties of each homology theory to show how various hypergraph properties imply properties of the homology groups. We also include discussion of functoriality for several of the homology theories. Finally, we provide a series of illustrative examples by computing many of these homology theories for small hypergraphs to show the variability of the methods and build intuition.

Figures (7)

• Figure 1: A diagram showing relationship between categories $\mathrm{\textbf{Simp}}$, $\mathrm{\textbf{Hyp}}$, and $\mathrm{\textbf{MHyp}}$ with $\hookrightarrow$ indicating full subcategory.
• Figure 2: Summary of relationships between a hypergraph and its dual, their upper closures, nerves, and closure homology.
• Figure 3: An illustration of (a) a hypergraph $\mathcal{H}$ transformed into (b) its upper closure $\Delta(\mathcal{H})$, (c) barycentric subdivision $\mathcal{B}(\Delta(\mathcal{H}))$, and (d) restricted barycentric subdivision $\mathcal{B}_{res}(\mathcal{H})$. Lower case letters represent vertices, whereas upper case letters represent hyperedges. Concatenations (e.g., ab, abc) in (c) and (d) represent simplices in $\Delta(\mathcal{H})$ such as edges and triangles.
• Figure 4: The polar complex for the example hypergraph in Figure \ref{['fig:rest_bary_ex']}(a) with representatives of the two $\mathsf{H}_{1}^{pol}(\mathcal{H})$ generators shown in pink and teal.
• Figure 5: Two hypergraphs to illustrate the fact that embedded homology captures "uniform cycles."

Theorems & Definitions (55)

• Definition 1
• Definition 2
• Definition 3: mac2013categories
• Definition 4: grilliette2023incidence
• Definition 5: green2025topological, page 7 with Def. 14; grilliette2020simplification, page 9 as $\mathrm{\textbf{SSys}}$
• Definition 6
• Proposition 7
• proof
• Proposition 8
• proof