用三种距离定义分析音乐图的拓扑结构，发现其同调结果存在包含关系。
Persistent Homology of Music Network with Three Different Distances
- 基于边路径定义三种不同距离度量音乐图节点
- 发现一维同调中持久条形码与图谱存在包含关系
- 适用于音乐数据拓扑分析与多视角结构挖掘
持久同调已被广泛应用于各类数据中隐藏拓扑结构的发现，包括音乐数据。应用持久同调需在点云或图网络节点间定义距离或度量，而此类定义不唯一，取决于具体问题目标。选择不同度量可导致多重拓扑推断。本文聚焦于预加权音乐图的持久同调应用，考察基于边路径的三种不同距离定义，展示其对持久条形码、持久图谱及出生/死亡边的影响。研究发现，这三种距离定义在1维持久同调中表现为条形码与图谱上的包含关系。该结论通过真实音乐数据验证。
原文摘要 · Abstract (English)
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.
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