Flagifying the Dowker Complex

2025-08-11

AI Summary

This work addresses the high computational complexity of Dowker complexes, which hinders their application in large-scale topological data analysis (TDA). We propose the Dowker–Rips complex—a lightweight alternative constructed via flagification. First formally defined, it strictly preserves the duality of the Dowker complex in 0- and 1-dimensional persistent homology. Moreover, we quantify the extent of duality failure in higher dimensions and elevate weak duality to the persistent homology level. Methodologically, our approach integrates flag complex construction, homological algebra, and interleaving distance theory, and we release an open-source Python implementation. Experiments on tumor microenvironment classification demonstrate substantial computational speedup—without sacrificing classification accuracy—enabling seamless integration into real-world TDA pipelines.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Other Foundations of Data Mining & Knowledge Management
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Web data integration and cleaning
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

The Dowker complex $mathrm{D}_{R}(X,Y)$ is a simplicial complex capturing the topological interplay between two finite sets $X$ and $Y$ under some relation $Rsubseteq X imes Y$. While its definition is asymmetric, the famous Dowker duality states that $mathrm{D}_{R}(X,Y)$ and $mathrm{D}_{R}(Y,X)$ have homotopy equivalent geometric realizations. We introduce the Dowker-Rips complex $mathrm{DR}_{R}(X,Y)$, defined as the flagification of the Dowker complex or, equivalently, as the maximal simplicial complex whose $1$-skeleton coincides with that of $mathrm{D}_{R}(X,Y)$. 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 $mathrm{H}_{i}(mathrm{DR}_{R}(X,Y))congmathrm{H}_{i}(mathrm{DR}_{R}(Y,X))$ 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.

Problem

• Develops Dowker-Rips complex for efficient topological data analysis.
• Explores weakened duality in homological dimensions for Dowker-Rips.
• Demonstrates computational speed gain in tumor classification pipeline.

Innovation

• Flagification of Dowker complex for efficiency
• Weakened duality holds for homology dimensions 0,1
• Python implementation enables faster classification pipelines