Flagifying the Dowker Complex

2025/08/11 by Marius Huber, Huber, Marius, Patrick Schnider +1 · 2 voices
Computer Science · Mathematics · #Algebraic Topology (math.AT) #Computational Geometry (cs.CG) #Digital Image Processing Techniques #FOS: Computer and information sciences #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #cs.CG #math.AT

paper · pdf · doi:10.48550/arxiv.2508.08025

openalex publication_date 2025/08/11 · arxiv published 2025/08/11 · arxiv updated 2025/08/19 · openalex created_date 2025/10/09 · openalex updated_date 2026/07/28

Abstract

The Dowker complex D_R(X,Y) is a simplicial complex capturing the topological interplay between two finite sets X and Y under some relation R⊆ X× Y. While its definition is asymmetric, the famous Dowker duality states that D_R(X,Y) and D_R(Y,X) have homotopy equivalent geometric realizations. We introduce the Dowker-Rips complex DR_R(X,Y), defined as the flagification of the Dowker complex or, equivalently, as the maximal simplicial complex whose 1-skeleton coincides with that of D_R(X,Y). This is motivated by applications in topological data analysis, since as a flag complex, the Dowker-Rips complex is less expensive to compute than the Dowker complex. While the Dowker duality does not hold for Dowker-Rips complexes in general, we show that one still has that H_i(DR_R(X,Y))\congH_i(DR_R(Y,X)) for i=0,1. We further show that this weakened duality extends to the setting of persistent homology, and quantify the ``failure" of the Dowker duality in homological dimensions higher than 1 by means of interleavings. This makes the Dowker-Rips complex a less expensive, approximate version of the Dowker complex that is usable in topological data analysis. Indeed, we provide a Python implementation of the Dowker-Rips complex and, as an application, we show that it can be used as a drop-in replacement for the Dowker complex in a tumor microenvironment classification pipeline. In that pipeline, using the Dowker-Rips complex leads to increase in speed while retaining classification performance.

Citations

• Relational persistent homology for multispecies data with application to the tumor microenvironment
• A functorial Dowker theorem and persistent homology of asymmetric networks
• Persistence stability for geometric complexes

Discussions

• A new paper on the #arXiv. Together with Marius Huber we are „Flagifying the Dowker Complex“ arxiv.org/abs/2508.08025 [bsky, 2 points, 1 comments]
• A new paper on the # arXiv . Together with Marius Huber we are „Flagifying the Dowker Complex“ https:// arxiv.org/abs/2508.08025 The Dowker complex is a simplicial complex used in # TDA to analyze the [mastodon, 0 points, 0 comments]

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