Homology, Homotopy and Applications

Volume 26 (2024)

Number 2

The magnitude homology of a hypergraph

Pages: 325 – 348

DOI: https://dx.doi.org/10.4310/HHA.2024.v26.n2.a16

Authors

Wanying Bi (School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, Hebei, China; and Beijing Institute of Mathematical Sciences and Applications, Beijing, China)

Jingyan Li (Beijing Institute of Mathematical Sciences and Applications, Beijing, China)

Jie Wu (Beijing Institute of Mathematical Sciences and Applications, Beijing, China)

Abstract

The magnitude homology, introduced by R. Hepworth and S. Willerton, offers a topological invariant that enables the study of graph properties. Hypergraphs, being a generalization of graphs, serve as popular mathematical models for data with higher-order structures. In this paper, we focus on describing the topological characteristics of hypergraphs by considering their magnitude homology. We begin by examining the distances between hyperedges in a hypergraph and establish the magnitude homology of hypergraphs. Additionally, we explore the relationship between the magnitude and the magnitude homology of hypergraphs. Furthermore, we derive several functorial properties of the magnitude homology for hypergraphs. Lastly, we present the Künneth theorem for the simple magnitude homology of hypergraphs.

Keywords

hypergraph, magnitude, magnitude homology, Künneth theorem

2010 Mathematics Subject Classification

05C65, 55N35, 55U25

Full Text (PDF format)

Received 15 August 2023

Accepted 8 November 2023

Published 9 October 2024