A complete characterization of the one-dimensional intrinsic Čech persistence diagrams for metric graphs.

*(English)*Zbl 1422.55037
Chambers, Erin Wolf (ed.) et al., Research in computational topology. Based on the first workshop for women in computational topology, Minneapolis, MN, USA, August 2016. Cham: Springer; Minneapolis, MN: Institute for Mathematics and its Applications (IMA). Assoc. Women Math. Ser. 13, 33-56 (2018).

This paper characterizes the 1-dimensional persistent homology of intrinsic Čech complexes of a metric graph \(G\) with a finite number of loops. Indeed, if the metric graph \(G\) is homotopy equivalent to a wedge sum of \(n\) circles, then the 1-dimensional persistent homology barcode has \(n\) intervals, each with birth time zero. The death times of the intervals are given by the lengths of a lexicographically shortest basis for 1-dimensional homology of \(G\) (divided by four). The algebraic arguments behind the proof are carefully constructed, and the figures illustrate the main ideas.

The result is well-motivated by applications of metric graphs to biology, to social networks, and to transportation systems. The paper has motivated extensions from metric graphs to geodesic spaces [Ž. Virk, Rev. Mat. Complut. 32, No. 1, 195–213 (2019; Zbl 1412.55018)]. The paper also advertises an interesting open question: the higher-dimensional persistent homology of a single loop is given in [M. Adamaszek and H. Adams, Pac. J. Math. 290, No. 1, 1–40 (2017; Zbl 1366.05124)], and the 1-dimensional persistent homology of general metric graphs is given in this paper, but what can be said about the higher-dimensional persistent homology of general metric graphs?

For the entire collection see [Zbl 1401.55001].

The result is well-motivated by applications of metric graphs to biology, to social networks, and to transportation systems. The paper has motivated extensions from metric graphs to geodesic spaces [Ž. Virk, Rev. Mat. Complut. 32, No. 1, 195–213 (2019; Zbl 1412.55018)]. The paper also advertises an interesting open question: the higher-dimensional persistent homology of a single loop is given in [M. Adamaszek and H. Adams, Pac. J. Math. 290, No. 1, 1–40 (2017; Zbl 1366.05124)], and the 1-dimensional persistent homology of general metric graphs is given in this paper, but what can be said about the higher-dimensional persistent homology of general metric graphs?

For the entire collection see [Zbl 1401.55001].

Reviewer: Henry Adams (Fort Collins)