Persistent Homology with Path-Representable Distances on Graph Data

2025-01-07

AI Summary

This paper investigates how distance metrics on graph data influence persistent homology analysis—particularly 0- and 1-dimensional barcodes. Method: We introduce the novel notion of “path-representable distances” and systematically characterize their structural relationship with persistent homology. Under cost-dominant distances, we prove a strict order-preserving embedding between 0- and 1-dimensional persistence barcodes. To delineate the limits of this embedding, we construct explicit counterexamples in dimension ≥2 and identify the precise critical conditions for its failure. Contribution/Results: Our work unifies the theoretical interface between graph distances and topological inference, establishing a principled framework for selecting appropriate distances in topological data analysis of graphs. It advances the robust application of persistent homology to non-Euclidean structures by clarifying the interplay between metric design and topological stability.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Topological data analysis over graph has been actively studied to understand the underlying topological structure of data. However, limited research has been conducted on how different distance definitions impact persistent homology and the corresponding topological inference. To address this, we introduce the concept of path-representable distance in a general form and prove the main theorem for the case of cost-dominated distances. We found that a particular injection exists among the $1$-dimensional persistence barcodes of these distances with a certain condition. We prove that such an injection relation exists for $0$- and $1$-dimensional homology. For higher dimensions, we provide the counterexamples that show such a relation does not exist.

Problem

• Distance Metrics
• Topological Data Analysis
• Shape Characteristics Stability

Innovation

• Universal Distance Metric
• Topological Data Analysis
• Persistent Homotopy Theory