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Persistent Homology of Music Network with Three Different Distances

Eunwoo Heo, Byeongchan Choi, Myung ock Kim, Mai Lan Tran, Jae-Hun Jung

TL;DR

This work investigates how persistent homology applied to music graphs depends on the chosen distance between graph nodes. It introduces three path-based distances (d1, d2, d3) on a weighted music network, proves that d2 is a metric while d1 and d3 may violate the triangle inequality, and establishes a universal inclusion d2 ≤ d3 ≤ d1 in both distances and one-dimensional PH barcodes. The authors prove injections between the corresponding birth sets and validate the theory with nine Korean traditional pieces plus a Western piece, demonstrating consistent birth times across distances and distance-specific death structures. The findings reveal a structured, non-random relationship among PH features under different distance definitions, with potential implications for musical analysis and AI-based composition. Overall, the study highlights how topological summaries of music can vary systematically with the underlying metric, offering multiple stylistic perspectives on the same musical 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.

Persistent Homology of Music Network with Three Different Distances

TL;DR

This work investigates how persistent homology applied to music graphs depends on the chosen distance between graph nodes. It introduces three path-based distances (d1, d2, d3) on a weighted music network, proves that d2 is a metric while d1 and d3 may violate the triangle inequality, and establishes a universal inclusion d2 ≤ d3 ≤ d1 in both distances and one-dimensional PH barcodes. The authors prove injections between the corresponding birth sets and validate the theory with nine Korean traditional pieces plus a Western piece, demonstrating consistent birth times across distances and distance-specific death structures. The findings reveal a structured, non-random relationship among PH features under different distance definitions, with potential implications for musical analysis and AI-based composition. Overall, the study highlights how topological summaries of music can vary systematically with the underlying metric, offering multiple stylistic perspectives on the same musical 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.

Paper Structure

This paper contains 10 sections, 7 theorems, 24 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Persistent homology on metric space
  3. Music graph networks and metric definitions
  4. A construction of music graph network
  5. Three path-based distance definitions
  6. Inequalities among three definitions
  7. Persistent homology induced by three different distances
  8. Injection relations among three different distances
  9. Examples
  10. Conclusion

Key Result

Proposition 3.7

Let $M=(V,E,W)$ be the music network and suppose that $W$ is positive. Denote $W^{-1} = \frac{1}{W}$ by the reciprocal weight function of $W$. If we define the reciprocal music network $M^{R}$ as $(V,E,W^{-1})$, then we have

Figures (13)

  • Figure 1: An example of filtration over a square point cloud on a plane.
  • Figure 2: Persistence barcode: zero dimension (left) and one dimension (right).
  • Figure 3: The persistence diagram of the zero (left) and one (right) dimensions for the unit square data example.
  • Figure 4: Distance between nodes. Left: the original graph $G$. Middle: $d(a,c)$ is defined by creating a new edge between $a$ and $c$. Right: $d(a,c)$ is defined based on the existing intermediate paths between $a$ and $c$.
  • Figure 5: A segment of the Korean traditional music piece Sanghyeondodeuri with five instruments.
  • ...and 8 more figures

Theorems & Definitions (19)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4: $d_1(v,w)$: The first definition of distance
  • Definition 3.5: $d_2(v,w)$: The second definition of distance
  • Definition 3.6: $d_3(v,w)$: The third definition of distance
  • Proposition 3.7
  • Proposition 3.8
  • proof
  • Example 3.9
  • ...and 9 more