The Persistent Homology of Cyclic Graphs

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