A survey of simplicial, relative, and chain complex homology theories for hypergraphs
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arXiv
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| Autori principali: | , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
| Accesso online: | |
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| _version_ | 1866912641429012480 |
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| author | Gasparovic, Ellen Purvine, Emilie Sazdanovic, Radmila Wang, Bei Wang, Yusu Ziegelmeier, Lori |
| author_facet | Gasparovic, Ellen Purvine, Emilie Sazdanovic, Radmila Wang, Bei Wang, Yusu Ziegelmeier, Lori |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18310 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A survey of simplicial, relative, and chain complex homology theories for hypergraphs Gasparovic, Ellen Purvine, Emilie Sazdanovic, Radmila Wang, Bei Wang, Yusu Ziegelmeier, Lori Algebraic Topology 55N35 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. |
| title | A survey of simplicial, relative, and chain complex homology theories for hypergraphs |
| topic | Algebraic Topology 55N35 |
| url | https://arxiv.org/abs/2409.18310 |