Barcode Embeddings for Metric Graphs

TL;DR

This paper investigates a homology-based invariant that embeds metric graphs into barcode space, proving its local and global injectivity properties, which enhances the discriminative power of topological invariants in applied topology.

Contribution

It establishes the local and global injectivity of the barcode embedding invariant for metric graphs, advancing understanding of their discriminative capabilities.

Findings

Invariant is locally injective on metric graphs

Invariant is globally injective on a GH-dense subset

Invariant is globally injective on a full measure subset

Abstract

Stable topological invariants are a cornerstone of persistence theory and applied topology, but their discriminative properties are often poorly-understood. In this paper we study a rich homology-based invariant first defined by Dey, Shi, and Wang, which we think of as embedding a metric graph in the barcode space. We prove that this invariant is locally injective on the space of metric graphs and globally injective on a GH-dense subset. Moreover, we show that is globally injective on a full measure subset of metric graphs, in the appropriate sense.