Discrete Morse theory, persistent homology and Forman–Ricci curvature

Using Banchoff’s discrete Morse theory, in tandem with Bloch’s result on the strong connection between the former and Forman’s Morse theory, and our own previous algorithm based on the later, we show that there exists a curvature-based, efficient persistent homology scheme for networks and hypernetworks. We also broaden the proposed method to include more general types of networks, by using Bloch’s extension of Banchoff’s work. Moreover, we show the connection between defect and Forman’s Ricci curvature that exists in the combinatorial setting, thus explaining previous empirical results showing very strong correlation between persistent homology results obtained using Forman’s Morse theory on the one hand, and Forman’s Ricci curvature, on the other.

Keywords

discrete Morse theory, persistent homology, Forman–Ricci curvature

2010 Mathematics Subject Classification

Primary 57Q99, 68R10. Secondary 05C82, 53Z99, 55U99.

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Research partially supported by the GIF Research Grant No. I-1514-304.6/2019.

Received 3 January 2021

Published 2 August 2022