↪︎ Monoidal Rips: Stable Multiparameter Filtrations of Directed Networks
Relaxing the metric assumption in topological data analysis allows for the analysis of directed networks and asymmetric data. The monoidal Rips filtration is a generalization of the Vietoris-Rips filtration to weighted directed graphs. To allow for directed edges, we work with filtered simplicial sets instead of filtered simplicial complexes. We also allow values in a class of lattices, which includes both the real numbers and products of totally ordered sets. This makes our construction applicable to both single-parameter and multiparameter persistence. We introduce a generalized network distance, and prove stability results for the persistent homology of the monoidal Rips filtration with respect to this distance.
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.
