Geometry-Induced Hodge Stars on Rips and Dowker--Rips Complexes

The Vietoris--Rips complex $\mathrm{VR}_\epsilon(X)$, the Dowker complex $\mathrm{D}_R(X,Y)$, and its flagified Dowker--Rips variant $\mathrm{DR}_R(X,Y)=\mathrm{F}(\mathrm{D}_R(X,Y))$ 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 $W_k=\operatorname{diag}\{w_k(\sigma):\sigma\in K_k\}$, which define weighted inner products on $k$-cochains. The resulting weighted discrete Hodge Laplacian $\Delta_k^W$ has kernel dimension equal to the $k$th 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 $s_t(\sigma;Y)$ 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 $\Delta_k^W$ for comparing geometry-aware Hodge spectra on Rips complexes.