# Hierarchical Clustering and Signal Denoising on Digraphs

arXiv: https://arxiv.org/abs/2609.34670 · HTML: https://mlsanity.com/paper/2609.34670/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 50 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Builds on (most cited first, 17 of 17)

- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [Multi-scale Attributed Node Embedding](https://mlsanity.com/paper/1909.13021/) (2019)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [MagNet: A Neural Network for Directed Graphs](https://mlsanity.com/paper/2102.11391/) (2021)
- [Convolutional Neural Network Architectures for Signals Supported on Graphs](https://mlsanity.com/paper/1805.00165/) (2018)
- [Graph Signal Processing: History, Development, Impact, and Outlook](https://mlsanity.com/paper/2303.12211/) (2023)
- [Clustering and Community Detection in Directed Networks: A Survey](https://mlsanity.com/paper/1308.0971/) (2013)
- [The Impact of Random Models on Clustering Similarity](https://mlsanity.com/paper/1701.06508/) (2017)
- [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)
- [DIGRAC: Digraph Clustering Based on Flow Imbalance](https://mlsanity.com/paper/2106.05194/) (2021)
- [Robust Graph Filter Identification and Graph Denoising from Signal Observations](https://mlsanity.com/paper/2210.08488/) (2022)
- [Localized Spectral Graph Filter Frames: A Unifying Framework, Survey of Design Considerations, and Numerical Comparison (Extended Cut)](https://mlsanity.com/paper/2006.11220/) (2020)
- [Semi-supervised Learning in Network-Structured Data via Total Variation Minimization](https://mlsanity.com/paper/1901.09838/) (2019)
- [Localized Linear Regression in Networked Data](https://mlsanity.com/paper/1903.11178/) (2019)
- [Generalized Sampling on Graphs With Subspace and Smoothness Priors](https://mlsanity.com/paper/1905.04441/) (2019)
- [Skew-Symmetric Adjacency Matrices for Clustering Directed Graphs](https://mlsanity.com/paper/2203.01388/) (2022)

## Abstract (arXiv)

In this paper, we propose a representation of a digraph (directed graph) as a Hermitian matrix derived from its adjacency matrix. This representation characterizes both the connectivity and the edge orientation of the digraph. Based on the spectral decomposition of the Hermitian matrix, a digraph clustering algorithm with k-means is introduced to produce a partition on the graph. Applying this algorithm (bottom-up) recursively to a digraph with partially labeled vertices yields a spectral hierarchical digraph clustering (myproj) algorithm that produces consistent nested partitions of the digraph, or equivalently, a tree structure. Furthermore, based on the in-degree and out-degree of each cluster in the digraph clustering, a pair of hierarchical interval partitions (filtrations) can be derived in a top-down manner to produce a pair of nested knot sequences. These knot sequences facilitate the construction of multilevel spline quasi-interpolants, enabling a noisy graph signal to be decomposed into a coarse approximation and inter-level details, followed by adaptive thresholding and reconstruction. Experiments on synthetic and real-world digraphs demonstrate the superiority of our myproj algorithm for digraph clustering across diverse graph structural properties (homophily and heterophily) and supervision settings. Moreover, experiments on digraph signal processing using multilevel spline quasi-interpolants further demonstrate the effectiveness of signal recovery on digraphs in terms of RMSE and SNR.