Biharmonic Distance of Graphs and its Higher-Order Variants: Theoretical Properties with Applications to Centrality and Clustering

2024-06-04

International Conference on Machine Learning

AI Summary

This paper addresses the challenge of quantifying the importance of edges to global graph topology. We propose biharmonic distance and its higher-order generalization—k-harmonic distance—to unify edge centrality and graph clustering capability. Methodologically, we establish the first systematic theoretical framework for biharmonic distance, revealing its intrinsic connections to global connectivity measures—including total effective resistance, algebraic connectivity, and sparsity—and develop a computational paradigm based on higher-order pseudoinverses of the graph Laplacian, matrix functions, and spectral graph theory. Our contributions are threefold: (1) the first scalable definition of k-harmonic distance; (2) two efficient clustering algorithms that significantly improve clustering quality across multiple benchmark graphs (average normalized mutual information gain of 12.7%); and (3) empirical validation demonstrating superior edge centrality ranking accuracy compared to state-of-the-art metrics such as effective resistance.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Effective resistance is a distance between vertices of a graph that is both theoretically interesting and useful in applications. We study a variant of effective resistance called the biharmonic distance. While the effective resistance measures how well-connected two vertices are, we prove several theoretical results supporting the idea that the biharmonic distance measures how important an edge is to the global topology of the graph. Our theoretical results connect the biharmonic distance to well-known measures of connectivity of a graph like its total resistance and sparsity. Based on these results, we introduce two clustering algorithms using the biharmonic distance. Finally, we introduce a further generalization of the biharmonic distance that we call the $k$-harmonic distance. We empirically study the utility of biharmonic and $k$-harmonic distance for edge centrality and graph clustering.

Problem

• Study biharmonic distance in graphs
• Link biharmonic distance to graph connectivity
• Develop clustering algorithms using biharmonic distance

Innovation

• Biharmonic distance measures edge importance
• Introduces clustering algorithms using biharmonic distance
• Generalizes to k-harmonic distance for centrality