Hierarchical Clustering and Signal Denoising on Digraphs

2026-09-28

AI Summary

This study addresses the challenges of directed graph clustering and signal denoising by proposing a spectral clustering framework that jointly accounts for connectivity and directionality. Methodologically, a Hermitian matrix is constructed to represent the directed graph structure, enabling hierarchical clustering through spectral decomposition combined with recursive K-means. Furthermore, the approach integrates hierarchical filtering with B-spline quasi-interpolation, achieving multiscale denoising and reconstruction of graph signals via adaptive thresholding. Experimental results on both synthetic and real-world datasets demonstrate that the proposed method significantly enhances clustering consistency while effectively improving signal recovery performance in terms of RMSE and SNR metrics.

Abstract

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.

Problem

• Directed Graphs
• Hierarchical Clustering
• Signal Denoising
• Graph Signal Processing

Innovation

• Directed Graphs
• Hermitian Matrix
• Spectral Hierarchical Clustering
• Multilevel Spline Quasi-interpolants
• Signal Denoising