  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.
