# The persistent homology of cyclic graphs

## TL;DR

This paper introduces a nearly quadratic time algorithm for computing the $k$-dimensional persistent homology of cyclic graph filtrations, significantly improving efficiency over traditional methods, especially for complexes derived from geometric data.

## Contribution

The authors develop an $O(n^2(k+ ext{log} n))$ algorithm for persistent homology of cyclic graph filtrations, applicable to Vietoris--Rips complexes from sampled points, with specific geometric conditions.

## Key findings

- Algorithm runs in nearly quadratic time in the number of vertices.
- Applicable to Vietoris--Rips complexes of points on curves in $ ext{R}^d$.
- Proven applicability for convex closed curves in $ ext{R}^2$ under certain conditions.

## Abstract

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.