Disentangling the Spectral Properties of the Hodge Laplacian: Not All Small Eigenvalues Are Equal

TL;DR

Not all small eigenvalues of the Hodge Laplacian $L_k$ carry the same information because harmonic, curl, and gradient components convey different geometric and topological structure. The authors define persistent eigenvector similarity (PES) and persistent eigenvector matching (PEM) to track harmonic, curl, and gradient eigenvectors across the $\alpha$-filtration, enabling a scale-aware analysis of the spectrum. They introduce Hodge spectral clustering and a region-edge metric via HGC-values, together with public code for reproducibility. The approach enhances interpretability and utility of higher-order spectral methods for simplicial complexes and point clouds, enabling topology-aware clustering and edge-role inference in complex networks.

Abstract

The rich spectral information of the graph Laplacian has been instrumental in graph theory, machine learning, and graph signal processing for applications such as graph classification, clustering, or eigenmode analysis. Recently, the Hodge Laplacian has come into focus as a generalisation of the ordinary Laplacian for higher-order graph models such as simplicial and cellular complexes. Akin to the traditional analysis of graph Laplacians, many authors analyse the smallest eigenvalues of the Hodge Laplacian, which are connected to important topological properties such as homology. However, small eigenvalues of the Hodge Laplacian can carry different information depending on whether they are related to curl or gradient eigenmodes, and thus may not be comparable. We therefore introduce the notion of persistent eigenvector similarity and provide a method to track individual harmonic, curl, and gradient eigenvectors/-values through the so-called persistence filtration, leveraging the full information contained in the Hodge-Laplacian spectrum across all possible scales of a point cloud. Finally, we use our insights (a) to introduce a novel form of Hodge spectral clustering and (b) to classify edges and higher-order simplices based on their relationship to the smallest harmonic, curl, and gradient eigenvectors.

Figures (4)

• Figure 1: Smallest eigenvectors at three stages of the $\alpha$-filtration.Blue: Harmonic, Green: Gradient, Red: Curl eigenvectors. The intensity of the colour denotes the absolute value of the corresponding entry in the eigenvector, where we scale the maximal entry to 1 for visualisation purposes. The three stages of the filtration correspond to the rows and represent three different regimes: 1st row: There are many small "holes" in the data set. Accordingly, the first five eigenvectors are harmonic and strongly localised. In the subsequent eigenvalues, gradient components dominate, which are being localised on the bridges between the clusters. The curl eigenvalues are large in comparison. 2nd row: There is a single harmonic eigenvector corresponding to the hole in the middle of the clusters, which is stronger on the bridges and the edges closer to the centre. The $\updownarrow$ and $\leftrightarrow$ gradient vectors are next and are localised at the bridges. We have a curl flow for each of the four clusters, and finally a diagonal gradient flow. 3rd row: There is a $1:1$ correspondence to the types of the previous vectors. However, the order of the eigenvectors has changed with the curl eigenvalues getting smaller. Both gradient and curl vectors are less localised.
• Figure 2: Evolution of the smallest eigenvalues of Hodge Laplacian across the steps of $\alpha$-filtration of the point cloud of \ref{['fig:eigenvectors']}.Top: We witness two distinct phases in eigenvalue behaviour. In the first regime of low filtration values, the associated $\alpha$-complexes are sparsely connected and exhibit many holes (large number of number of harmonic eigenvalues $\lambda=0$). There is a V-shaped pattern of gradient eigenvalues. This correspond to the fact that the clusters first grow, which creates very "thin" bridges (small gradient eigenvalues) between growing clusters. Once the clusters start to grow together the connecting bridges become "wider" (more connecting edges), and the gradient eigenvalues increase again. In the second regime of medium to high filtration values, the associated $\alpha$-complexes grow denser and the holes are filled. Only one harmonic eigenvalue remains, whereas three structural gradient eigenvalues remain relatively small. However, the most distinct feature of the second regime is the cohort of decreasing curl eigenvectors/-values overtaking the gradient vectors of the initial V-shape. The smallest four of these eigenvalues are separated from the rest by a significant margin. They represent the curl flow generated around each of the four clusters. Bottom: Number of eigenvalues in smallest $40$ being curl, gradient, and harmonic.
• Figure 3: Hodge Spectral Clustering. In contrast to ordinary $0$-dimensional spectral clustering, higher-order spectral clustering allows for the subcategories of gradient, curl and harmonic clustering, highlighting different features of the simplicial complex.
• Figure 4: Edge roles using HGC-values. Relative strength of blue: harmonic, red: curl, green: gradient contributions. Top:$\alpha$-complexes on two 3d point clouds sampled on real world objects of ModelNet40 data set wu20153d. Bottom:$\alpha$-complexes on two synthetic point clouds.

Theorems & Definitions (7)

• Definition 2.1: Simplicial complex
• Definition 2.2: Hodge Laplacian
• Theorem 2.3: Hodge Decomposition Lim:2020Barbarossa:2020Schaub:2021Roddenberry:2021
• Definition 2.4: $\alpha$-complexes
• Definition 3.1: Persistent eigenvector similarity
• Definition 3.2: Persistent eigenvector matching (PEM)
• Definition 4.1: HGC-values