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

Hypergraphs have seen widespread applications in network and data science communities in recent years. We present a survey of recent work to define topological objects from hypergraphs -- specifically simplicial, relative, and chain complexes -- that can be used to build homology theories for hypergraphs. We define and describe nine different homology theories and their relevant topological objects. We discuss some interesting properties of each method to show how the hypergraph structures are preserved or destroyed by modifying a hypergraph. 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.