Persistent Homology of Music Network with Three Different Distances

TL;DR

This paper explores how different distance definitions in music graphs influence persistent homology analysis, revealing inclusion relations among topological features and demonstrating the impact on persistence diagrams with real music data.

Contribution

It introduces three distinct distance definitions for music graphs and analyzes their effects on persistent homology, highlighting inclusion relations and their implications.

Findings

Inclusion relations exist among the persistent homology features derived from the three distance definitions.

Different distance metrics significantly affect the resulting persistence diagrams and barcodes.

The findings are validated using real-world music data.

Abstract

Persistent homology has been widely used to discover hidden topological structures in data across various applications, including music data. To apply persistent homology, a distance or metric must be defined between points in a point cloud or between nodes in a graph network. These definitions are not unique and depend on the specific objectives of a given problem. In other words, selecting different metric definitions allows for multiple topological inferences. In this work, we focus on applying persistent homology to music graph with predefined weights. We examine three distinct distance definitions based on edge-wise pathways and demonstrate how these definitions affect persistent barcodes, persistence diagrams, and birth/death edges. We found that there exist inclusion relations in one-dimensional persistent homology reflected on persistence barcode and diagram among these three distance definitions. We verified these findings using real music data.