Asian Journal of Mathematics

Volume 23 (2019)

Number 3

The embedded homology of hypergraphs and applications

Pages: 479 – 500

DOI: https://dx.doi.org/10.4310/AJM.2019.v23.n3.a6

Authors

Stephane Bressan (School of Computing, National University of Singapore)

Jingyan Li (Department of Mathematics and Physics, Shijiazhuang Tiedao University, China)

Shiquan Ren (Yau Mathematical Sciences Center, Tsinghua University, China)

Jie Wu (College of Mathematics and Information Science, Hebei Normal University, China wujie@hebtu.edu.cn)

Abstract

Hypergraphs are mathematical models for many problems in data sciences. In recent decades, the topological properties of hypergraphs have been studied and various kinds of (co)homologies have been constructed (cf. [4, 7, 19]). In this paper, generalising the usual homology of simplicial complexes, we define the embedded homology of hypergraphs as well as the persistent embedded homology of sequences of hypergraphs. As a generalisation of the Mayer–Vietoris sequence for the homology of simplicial complexes, we give a Mayer–Vietoris sequence for the embedded homology of hypergraphs. Moreover, as applications of the embedded homology, we study acyclic hypergraphs and construct some indices for the data analysis of hyper-networks.

Keywords

hypergraph, acyclic hypergraph, homology, persistent homology, Mayer–Vietoris sequence, hyper-network

2010 Mathematics Subject Classification

Primary 55U10, 55U15. Secondary 68P05, 68P15.

Full Text (PDF format)

This work was supported by the National Research Foundation, Prime Minister’s Office, Singapore under its Campus for Research Excellence and Technological Enterprise (CREATE) programme.

Received 4 October 2016

Accepted 14 February 2018

Published 9 July 2019