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Monoidal Rips: Stable Multiparameter Filtrations of Directed Networks

Nello Blaser, Morten Brun, Odin Hoff Gardaa, Lars M. Salbu

TL;DR

The paper develops the monoidal Rips filtration for $L$-graphs valued in a symmetric duoidal lattice, unifying and extending directed and metric-based persistent constructions by replacing the max with a monoidal product $\otimes$. It proves stability results for the resulting persistence modules using a generalized network distance $d_\mathcal{N}^\oplus$, and extends the theory to multiparameter persistence when $L$ is a product of totally ordered lattices, including a novel stability bound for the sublevel Rips bifiltration. The authors implement the $p$-Rips filtration for $[0,\infty]$-graphs, demonstrate competitive or superior performance to Flagser in graph regression tasks, and show that combining different monoidal products can improve classification in point-cloud experiments. This framework provides a flexible, stable approach to topological analysis of directed networks and multiparameter datasets, with practical impact on network analysis and geometry-informed learning.

Abstract

We introduce the monoidal Rips filtration, a filtered simplicial set for weighted directed graphs and other lattice-valued networks. Our construction generalizes the Vietoris-Rips filtration for metric spaces by replacing the maximum operator, determining the filtration values, with a more general monoidal product. We establish interleaving guarantees for the monoidal Rips persistent homology, capturing existing stability results for real-valued networks. When the lattice is a product of totally ordered sets, we are in the setting of multiparameter persistence. Here, the interleaving distance is bounded in terms of a generalized network distance. We use this to prove a novel stability result for the sublevel Rips bifiltration. Our experimental results show that our method performs better than Flagser in a graph regression task, and that combining different monoidal products in point cloud classification can improve performance.

Monoidal Rips: Stable Multiparameter Filtrations of Directed Networks

TL;DR

The paper develops the monoidal Rips filtration for -graphs valued in a symmetric duoidal lattice, unifying and extending directed and metric-based persistent constructions by replacing the max with a monoidal product . It proves stability results for the resulting persistence modules using a generalized network distance , and extends the theory to multiparameter persistence when is a product of totally ordered lattices, including a novel stability bound for the sublevel Rips bifiltration. The authors implement the -Rips filtration for -graphs, demonstrate competitive or superior performance to Flagser in graph regression tasks, and show that combining different monoidal products can improve classification in point-cloud experiments. This framework provides a flexible, stable approach to topological analysis of directed networks and multiparameter datasets, with practical impact on network analysis and geometry-informed learning.

Abstract

We introduce the monoidal Rips filtration, a filtered simplicial set for weighted directed graphs and other lattice-valued networks. Our construction generalizes the Vietoris-Rips filtration for metric spaces by replacing the maximum operator, determining the filtration values, with a more general monoidal product. We establish interleaving guarantees for the monoidal Rips persistent homology, capturing existing stability results for real-valued networks. When the lattice is a product of totally ordered sets, we are in the setting of multiparameter persistence. Here, the interleaving distance is bounded in terms of a generalized network distance. We use this to prove a novel stability result for the sublevel Rips bifiltration. Our experimental results show that our method performs better than Flagser in a graph regression task, and that combining different monoidal products in point cloud classification can improve performance.

Paper Structure

This paper contains 25 sections, 22 theorems, 66 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symmetric Monoidal- and Duoidal Lattices
  4. Galois Connections and Adjunctions
  5. Persistence Modules and Interleavings
  6. Simplicial Sets
  7. L-graphs and the Monoidal Rips Filtration
  8. The Graph Distance
  9. Stability
  10. Surjections Induce Homotopy Equivalences
  11. Main Stability Results
  12. Multipersistence and the Generalized Network Distance
  13. The Generalized Network Distance
  14. Multipersistence Stability
  15. Example: The Sublevel Rips Bifiltration
  16. ...and 10 more sections

Key Result

Corollary 1.0

Let $(L,\leq,\otimes,\oplus,e)$ be a symmetric duoidal lattice. For any two $L$-graphs $G,G'\colon V\times V\to L$ and $n\geq 0$, the corresponding monoidal Rips persistence modules $H_n(R_\otimes^\bullet(G))$ and $H_n(R_\otimes^\bullet(G'))$ are $\delta_\oplus(G,G')^{\otimes (n+1)}$-interleaved.

Figures (2)

  • Figure 1: Box plot showing mean absolute errors (MAE) for our filtration and Flagser over the $100$ experiments in our geometric graph regression problem. Note that outliers are not shown in the plot. The dashed line $x=1.25$ indicates the expected MAE for the constant function predicting $\hat{\alpha}=7.5$.
  • Figure 2: Box plot showing accuracies for the different values of $p$ over $100$ experiments in our point cloud classification problem. Note that outliers are not shown in the plot.

Theorems & Definitions (63)

  • Corollary 1.0: Graph Distance Stability
  • Theorem 1.0: Correspondence Stability
  • Theorem 1.0: Generalized Network Distance Stability
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Example 2.6
  • ...and 53 more