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

arXiv: https://arxiv.org/abs/2203.03221 · HTML: https://mlsanity.com/paper/2203.03221/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 31 references that are in the map.
- First posted on arXiv 7 Mar 2022.

## Builds on (most cited first, 7 of 7)

- [Hermitian matrices for clustering directed graphs: insights and applications](https://mlsanity.com/paper/1908.02096/) (2019)
- [Higher-Order Spectral Clustering of Directed Graphs](https://mlsanity.com/paper/2011.05080/) (2020)
- [Multiscale dynamical embeddings of complex networks](https://mlsanity.com/paper/1804.03733/) (2018)
- [Semi-supervised classification on graphs using explicit diffusion dynamics](https://mlsanity.com/paper/1909.11117/) (2019)
- [Skew-Symmetric Adjacency Matrices for Clustering Directed Graphs](https://mlsanity.com/paper/2203.01388/) (2022)
- [A Performance Guarantee for Spectral Clustering](https://mlsanity.com/paper/2007.05627/) (2020)
- [A simpler spectral approach for clustering in directed networks](https://mlsanity.com/paper/2102.03188/) (2021)

## Built on by (most cited first, 3 of 3)

- [Parametrized Power-Iteration Clustering for Directed Graphs](https://mlsanity.com/paper/2210.00310/) (2022)
- [Eigenvalue-based Incremental Spectral Clustering](https://mlsanity.com/paper/2308.10999/) (2023)
- [Multi-view diffusion geometry using intertwined diffusion trajectories](https://mlsanity.com/paper/2512.01484/) (2025)

## Abstract (arXiv)

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.