Distances and Isomorphism between Networks: Stability and Convergence of Network Invariants

TL;DR

This paper establishes a theoretical framework for a generalized Gromov-Hausdorff distance on networks, enabling continuum limits, stability analysis, and well-defined persistent homology in the infinite setting, with applications to network invariants.

Contribution

It introduces a continuum limit for finite networks, characterizes network isomorphisms, and develops a unified, stable framework for network invariants including persistent homology.

Findings

Defined continuum limits for finite networks.

Characterized network isomorphisms in the infinite setting.

Demonstrated stability of persistent homology methods.

Abstract

We develop the theoretical foundations of a generalized Gromov-Hausdorff distance between functions on networks that has recently been applied to various subfields of topological data analysis and optimal transport. These functional representations of networks, or networks for short, specialize in the finite setting to (possibly asymmetric) adjacency matrices and derived representations such as distance or kernel matrices. Existing literature utilizing these constructions cannot, however, benefit from continuous formulations because the continuum limits of finite networks under this distance are not well-understood. For example, while there are currently numerous persistent homology methods on finite networks, it is unclear if these methods produce well-defined persistence diagrams in the infinite setting. We resolve this situation by introducing the collection of compact networks that arises by taking continuum limits of finite networks and developing sampling results showing that this collection admits well-defined persistence diagrams. Compared to metric spaces, the isomorphism class of the generalized Gromov-Hausdorff distance over networks is rather complex, and contains representatives having different cardinalities and different topologies. We provide an exact characterization of a suitable notion of isomorphism for compact networks as well as alternative, stronger characterizations under additional topological regularity assumptions. Toward data applications, we describe a unified framework for developing quantitatively stable network invariants, provide basic examples, and cast existing results on the stability of persistent homology methods in this extended framework. To illustrate our theoretical results, we introduce a model of directed circles with finite reversibility and characterize their Dowker persistence diagrams.