2025/08/11 by Huber, Marius, Schnider, Patrick
#Algebraic Topology (math.AT) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics
paper · doi:10.48550/arxiv.2508.08025
The Dowker complex DR(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 DR(X,Y) and DR(Y,X) have homotopy equivalent geometric realizations. We introduce the Dowker-Rips complex DRR(X,Y), defined as the flagification of the Dowker complex or, equivalently, as the maximal simplicial complex whose 1-skeleton coincides with that of DR(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 Hi(DRR(X,Y))\congHi(DRR(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.