Generalized Dirichlet Energy and Graph Laplacians for Clustering Directed and Undirected Graphs

TL;DR

This paper introduces the generalized Dirichlet energy (GDE) and a new spectral clustering method that effectively handles both directed and undirected graphs without losing directional information, improving clustering accuracy.

Contribution

The work presents a unified GDE framework and a generalized spectral clustering method that directly incorporates directionality, avoiding symmetrization and teleportation tricks used in prior methods.

Findings

GSC outperforms existing spectral clustering methods in accuracy

GDE effectively captures directionality and density in graphs

Method demonstrates robustness on real-world datasets

Abstract

Clustering in directed graphs remains a fundamental challenge due to the asymmetry in edge connectivity, which limits the applicability of classical spectral methods originally designed for undirected graphs. A common workaround is to symmetrize the adjacency matrix, but this often leads to losing critical directional information. In this work, we introduce the generalized Dirichlet energy (GDE), a novel energy functional that extends the classical Dirichlet energy to handle arbitrary positive vertex measures and Markov transition matrices. GDE provides a unified framework applicable to both directed and undirected graphs, and is closely tied to the diffusion dynamics of random walks. Building on this framework, we propose the generalized spectral clustering (GSC) method that enables the principled clustering of weakly connected digraphs without resorting to the introduction of teleportation to the random walk transition matrix. A key component of our approach is the utilization of a parametrized vertex measure encoding graph directionality and density. Experiments on real-world point-cloud datasets demonstrate that GSC consistently outperforms existing spectral clustering approaches in terms of clustering accuracy and robustness, offering a powerful new tool for graph-based data analysis.