Persistent Homology of Music Network with Three Different Distances

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.