Persistent Homology of Music Network with Three Different Distances

2025-06-16

AI Summary

Selecting an appropriate graph distance metric is critical for topological analysis of musical networks via persistent homology, yet the impact of different distance definitions on homological outputs remains poorly understood. Method: We systematically compute one-dimensional persistent barcodes and persistence diagrams on weighted musical graphs using three distinct graph distance definitions: shortest-path distance, weighted-path distance, and cumulative similarity distance. Contribution/Results: We establish, for the first time, a strict inclusion relationship among the three distance-induced persistence modules at the one-dimensional homology level: their barcodes exhibit a nested structure, enabling precise characterization of the hierarchical scale of topological features. This yields a theoretically grounded, interpretable criterion for distance selection in musical network analysis. Experiments on real-world musical datasets confirm the robustness of this hierarchy, significantly enhancing the interpretability and stability of higher-order structural representations. Our findings broaden the theoretical and practical foundations of persistent homology in complex musical networks.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Social media analysis through the lenses of networks

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.

Problem

• Explore topological structures in music networks using persistent homology
• Compare three distance definitions' impact on persistent barcode outputs
• Analyze inclusion relations in persistence diagrams from music data

Innovation

• Applied persistent homology to music graph
• Examined three distinct distance definitions
• Verified findings with real music data