The Persistent Homology of Cyclic Graphs

Abstract

We give an [Formula: see text] algorithm for computing the [Formula: see text]-dimensional persistent homology of a filtration of clique complexes of cyclic graphs on [Formula: see text] vertices. This is nearly quadratic in the number of vertices [Formula: see text], and therefore a large improvement upon the traditional persistent homology algorithm, which is cubic in the number of simplices of dimension at most [Formula: see text], and hence of running time [Formula: see text] in the number of vertices [Formula: see text]. Our algorithm applies, for example, to Vietoris–Rips complexes of points sampled from a curve in [Formula: see text] when the scale is bounded depending on the geometry of the curve, but still large enough so that the Vietoris–Rips complex may have non-trivial homology in arbitrarily high dimensions [Formula: see text]. In the case of the plane [Formula: see text], we prove that our algorithm applies for all scale parameters if the [Formula: see text] vertices are sampled from a convex closed differentiable curve whose convex hull contains its evolute. We ask if there are other geometric settings in which computing persistent homology is (say) quadratic or cubic in the number of vertices, instead of in the number of simplices.