2026/07/21 by Jiahui Chen, Sebastian Wilcox
#math.AT
The Vietoris--Rips complex VRε(X), the Dowker complex DR(X,Y), and its flagified Dowker--Rips variant DRR(X,Y)=F(DR(X,Y)) are simplicial complexes constructed from metric data or witness relations. They are useful in topological data analysis because they encode topology through combinatorial data derived from pairwise information, but at a fixed scale they retain little of the underlying geometry. Unlike alpha complexes or mesh-based discretizations, Rips-type complexes carry no canonical primal--dual cell structure, which is the ingredient used by the discrete exterior calculus Hodge star to encode metric information. We address this gap by equipping a Rips-type complex K with diagonal geometry-induced Hodge stars represented by positive simplex weights Wk=diag\wk(σ):σ∈ Kk\, which define weighted inner products on k-cochains. The resulting weighted discrete Hodge Laplacian ΔkW has kernel dimension equal to the kth Betti number of the underlying complex, while its nonzero spectrum is governed by the chosen geometric weights. The central issue is therefore not the existence of a weighted Laplacian, since any positive diagonal weights define one, but the design of weights that encode meaningful metric or witness geometry. We focus on two computable choices: simplex-volume weights, based on Euclidean simplex volumes, and soft witness weights, based on a Dowker-style support function st(σ;Y) that quantifies higher-order witness support lost under flagification. We prove positivity, weighted self-adjointness, and Betti-number preservation for arbitrary positive diagonal weights, establish an asymptotic decay-rate characterization for soft witness support, and describe spectral descriptors derived from ΔkW for comparing geometry-aware Hodge spectra on Rips complexes.