The persistent homology of cyclic graphs

We give an $O(n^2(k+\log n))$ algorithm for computing the $k$-dimensional persistent homology of a filtration of clique complexes of cyclic graphs on $n$ vertices. This is nearly quadratic in the number of vertices $n$, and therefore a large improvement upon the traditional persistent homology algorithm, which is cubic in the number of simplices of dimension at most $k+1$, and hence of running time $O(n^{3(k+2)})$ in the number of vertices $n$. Our algorithm applies, for example, to Vietoris--Rips complexes of points sampled from a curve in $\mathbb{R}^d$ 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 $k$. In the case of the plane $\mathbb{R}^2$, we prove that our algorithm applies for all scale parameters if the $n$ 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.