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

arXiv: https://arxiv.org/abs/2311.14427 · HTML: https://mlsanity.com/paper/2311.14427/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 27 references that are in the map.
- First posted on arXiv 24 Nov 2023.

## Builds on (most cited first, 13 of 13)

- [Topological Signal Processing over Simplicial Complexes](https://mlsanity.com/paper/1907.11577/) (2019)
- [Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond](https://mlsanity.com/paper/2101.05510/) (2021)
- [Principled Simplicial Neural Networks for Trajectory Prediction](https://mlsanity.com/paper/2102.10058/) (2021)
- [Persistent Laplacians: properties, algorithms and implications](https://mlsanity.com/paper/2012.02808/) (2020)
- [Edges in Brain Networks: Contributions to Models of Structure and Function](https://mlsanity.com/paper/2105.07069/) (2021)
- [Approximation Algorithms for Reducing the Spectral Radius to Control Epidemic Spread](https://mlsanity.com/paper/1501.06614/) (2015)
- [A Notion of Harmonic Clustering in Simplicial Complexes](https://mlsanity.com/paper/1910.07247/) (2019)
- [Spectral Detection of Simplicial Communities via Hodge Laplacians](https://mlsanity.com/paper/2108.06547/) (2021)
- [Topological Point Cloud Clustering](https://mlsanity.com/paper/2303.16716/) (2023)
- [Outlier Detection for Trajectories via Flow-embeddings](https://mlsanity.com/paper/2111.13235/) (2021)
- [Structure of complex networks: Quantifying edge-to-edge relations by failure-induced flow redistribution](https://mlsanity.com/paper/1303.6241/) (2013)
- [A Novel Measure of Edge Centrality in Social Networks](https://mlsanity.com/paper/1303.1747/) (2013)
- [Clustering with Simplicial Complexes](https://mlsanity.com/paper/2303.07646/) (2023)

## Built on by (most cited first, 5 of 5)

- [Topological Trajectory Classification and Landmark Inference on Simplicial Complexes](https://mlsanity.com/paper/2412.03145/) (2024)
- [HodgeCover: Higher-Order Topological Coverage Drives Compression of Sparse Mixture-of-Experts](https://mlsanity.com/paper/2605.13997/) (2026)
- [A Spectral Framework for Tracking Communities in Evolving Networks](https://mlsanity.com/paper/2412.07378/) (2024)
- [Topological Signal Processing: An Application-Oriented Tutorial](https://mlsanity.com/paper/2605.22853/) (2026)
- [Selecting Interpretable Circular Coordinates from Data](https://mlsanity.com/paper/2607.08230/) (2026)

## Abstract (arXiv)

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.