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Flagifying the Dowker Complex
Marius Huber, Patrick Schnider
TL;DR
The paper addresses whether a cheaper flagified approximation can preserve the useful duality and topology of Dowker complexes. It introduces Dowker-Rips complexes, proves interleaving and low-dimensional persistent-homology guarantees, and shows an application with faster computation and retained classification performance.
Problem
The paper asks how much Dowker-Rips complexes can differ from Dowker complexes and whether Dowker duality persists after flagification.
Source: §1 · p. 3Method
The paper defines Dowker-Rips complexes by flagifying Dowker complexes and studies their metric filtrations using multiplicative interleavings and homological comparisons.
Source: §1 · pp. 2–3Results
Replacing the Dowker complex with the Dowker-Rips complex increases speed without negatively affecting classification performance, while persistent-homology duality holds in dimensions i = 0, 1.
Source: §1 · p. 4Takeaways & Limitations
Dowker-Rips complexes provide a less expensive approximate alternative that retains low-dimensional duality and can function as a practical replacement in the reported classification pipeline.
Source: §1 · pp. 2, 4Persistent-homology isomorphism under swapping the two sets does not generally extend beyond homological dimensions 0 and 1.
Source: §1 · p. 3
Abstract
from arXiv · showhide
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 $R\subseteq X\times Y$. 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 $\mathrm{H}_{i}(\mathrm{DR}_{R}(X,Y))\cong\mathrm{H}_{i}(\mathrm{DR}_{R}(Y,X))$ 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.
1 Introduction
The paper introduces the Dowker-Rips complex as a computationally cheaper flagified approximation to the Dowker complex, then studies how much duality and topological similarity survive. It proves multiplicative interleaving guarantees, low-dimensional persistent-homology duality, and practical speed gains without reduced classification performance.
1 Introduction
- Dowker complexes encode relations between two sets, with simplices in DR(X,Y) consisting of subsets of X jointly related to some element of Y.For metric subsets, the relation is defined by d(x,y) ≤ ε, making the complex a relative analogue of the Čech complex.Source: §1 · p. 2
- The Dowker-Rips complex is introduced as the flagification of the Dowker complex, retaining its 1-skeleton while adding simplices supported by existing edges.This mirrors the Vietoris-Rips approximation to the Čech complex and is intended to reduce computation and storage costs.Source: §1 · pp. 1–2
- The paper asks how Dowker-Rips complexes differ from Dowker complexes and whether Dowker duality persists after flagification.These questions motivate interleaving bounds and homological comparisons between the two constructions.Source: §1 · p. 3
- For metric relations, the Dowker-Rips and Dowker filtrations are multiplicatively 3-interleaved, and this factor is sharp.The inclusion DRε(X,Y) ⊆ D3ε(X,Y) can fail for every replacement factor c < 3.Source: §1 · p. 3
- The filtrations DR•(X,Y) and DR•(Y,X) are also multiplicatively 3-interleaved, providing an approximation to Dowker duality rather than a full equivalence.Unlike classical Dowker duality, this guarantee is weaker than persistent-homology isomorphism in every dimension.Source: §1 · p. 3
- Dowker-Rips persistent homology is isomorphic under swapping X and Y in dimensions i = 0, 1, but this conclusion fails in higher dimensions.Because persistent homology is often computed only through dimension 1, the duality can allow choosing the less expensive of the two complexes.Source: §1 · pp. 3–4
- Replacing the Dowker complex with the Dowker-Rips complex in a tumor microenvironment classification pipeline increases speed without negatively affecting classification performance.The paper presents this application as evidence that the flagified complex can serve as a practical drop-in replacement.Source: §1 · p. 4
2 Multiplicative interleavings
The paper establishes multiplicative interleavings between Dowker and Dowker-Rips filtrations, and between the two Dowker-Rips filtrations under metric relations. The factor 3 is sharp for the Dowker-to-Dowker-Rips comparison.
2 Multiplicative interleavings
- Multiplicative interleavings quantify similarity between filtrations when additive interleavings are not the appropriate guarantee.The paper situates its factor-3 results alongside the multiplicative 2-interleaving between Čech and Vietoris-Rips filtrations.Source: §2 · pp. 4–5
- The Dowker and Dowker-Rips filtrations are multiplicatively 3-interleaved for metric relations R_ε defined by d(x,y) ≤ ε.The forward maps are inclusions, and the required diagrams commute accordingly.Source: §2 · p. 5
- The two Dowker-Rips filtrations on (X,Y) and (Y,X) are multiplicatively 3-interleaved, using general interleaving maps rather than only inclusion chains.The construction relies on maps and commuting diagrams, with functoriality of Γ ensuring commutativity.Source: §2 · pp. 5–6
- The Dowker-to-Dowker-Rips factor 3 is sharp: the inclusion DR_ε(X,Y) ⊆ D_cε(X,Y) can fail for every c < 3.A graph-metric example has a triangle in DR_1(X,Y), while a corresponding Dowker simplex requires c ≥ 3.Source: §2 · p. 6
3 Dowker-Rips duality
The paper extends Dowker duality to flagified Dowker-Rips complexes through partial flagifications, obtaining homology isomorphisms below dimension k and persistence-module isomorphisms in corresponding dimensions. For k=2, this gives persistent duality in dimensions 0 and 1, while an explicit example shows failure in dimension 2.
Proof strategy
- The proof framework begins with the simplicial map Γ underlying classical Dowker duality and extends it from the barycentric subdivision to partial flagifications.The extension is defined simplex by simplex, while repeated choices can be made consistently across inclusions.Source: §3 · pp. 7, 8, 10
Proof mechanism
- The construction fills each newly added k-simplex by building a corresponding contractible complex on the dual side.The proof uses barycentric subdivisions, shared-intersection simplices, and iterative collapses to a k-simplex.Source: §3 · pp. 7–8
Technical construction
- Partial k-flagifications extend the Dowker homotopy equivalence while preserving homology through dimension k −1 and compatibility across relation inclusions.The extensions are constructed so the relevant diagrams commute up to homotopy, with precise commutativity obtainable by consistent choices.Source: §3 · pp. 8–10
Main theorem
- Persistent homology modules of k-flagified Dowker-Rips filtrations are isomorphic in dimensions i = 0, . . . , k −1.The result applies to increasing sequences of relations and follows from commuting maps between partial flagifications and filtration inclusions.Source: §3 · pp. 10–11
Dowker-Rips duality
- For k = 2, the Dowker-Rips filtrations have isomorphic persistent homology in dimensions 0 and 1.This is obtained by specializing the general filtration theorem to k = 2 for metric relations Rε defined by distance thresholds.Source: §3 · p. 11
Sharpness
- The dimension bound is sharp: for a regular tetrahedron and edge-midpoint relation, the two flagifications are homotopy equivalent to a point and a 2-sphere.Thus the two complexes have different second homology groups in the exhibited filtration level.Source: §3 · p. 11
4 The Dowker-Rips complex as a drop-in replacement for the Dowker complex
Replacing the Dowker complex with the Dowker-Rips complex in a tumor microenvironment classification pipeline preserved classification performance while substantially reducing computation time.
- The pipeline represents labeled microenvironments as two-dimensional point clouds, computes three classes of persistent-homology features, and classifies them with an SVM.The three pairings are macrophage–tumor, tumor–blood vessel, and macrophage–blood vessel.Source: §4 · p. 12
- Over 14× faster computation of the relevant complexes and persistent homologies was achieved with the Dowker-Rips replacement, while classification performance remained essentially unchanged.The comparison used average accuracy, standard deviation, and median accuracy across ten runs.Source: §4 · p. 12
- Dowker-Rips complexes therefore function as a drop-in replacement in this classification workflow without negatively affecting its reported predictive performance.Source: §4 · p. 12