Topological Study of β-Sparsified d-Uniform Hypergraph-Based Simplicial Complexes

Persistent Homology (PH) is a method of Topological Data Analysis that characterizes the topological structure of a space. Unfortunately, the computation of PH for high-dimensional and big data is not possible due to the exponential growth of the constructed complex. Fortunately, sparsification techniques can substantially reduce the size of the complex. This paper examines a sparsification technique (β-Sparsification) that produces a complex reduction capability that is scalable to a user-specified value β. At β=0 this scaling generates complexes that can have the same 1-Skeleton as the Vietoris–Rips complex; β=1 produces a Delaunay complex, and other values of β produce a range of (unnamed) complexes. Experiments with β-Sparsification reveal that the topology of the sparsified simplicial complex is preserved for 0≤β≤1; for β>1, the complex begins to lose (potentially insignificant) topological features.