# Hypergraphs with Edge-Dependent Vertex Weights: Spectral Clustering based on the 1-Laplacian

arXiv: https://arxiv.org/abs/2305.00462 · HTML: https://mlsanity.com/paper/2305.00462/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 26 references that are in the map.
- First posted on arXiv 30 Apr 2023.

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

- [Networks beyond pairwise interactions: structure and dynamics](https://mlsanity.com/paper/2006.01764/) (2020)
- [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)
- [Random Walks on Hypergraphs with Edge-Dependent Vertex Weights](https://mlsanity.com/paper/1905.08287/) (2019)
- [The Total Variation on Hypergraphs - Learning on Hypergraphs Revisited](https://mlsanity.com/paper/1312.5179/) (2013)
- [Inhomogeneous Hypergraph Clustering with Applications](https://mlsanity.com/paper/1709.01249/) (2017)
- [Submodular Hypergraphs: p-Laplacians, Cheeger Inequalities and Spectral Clustering](https://mlsanity.com/paper/1803.03833/) (2018)
- [Hypergraph Random Walks, Laplacians, and Clustering](https://mlsanity.com/paper/2006.16377/) (2020)
- [Hypergraph Cuts with General Splitting Functions](https://mlsanity.com/paper/2001.02817/) (2020)
- [Cheeger Inequalities for Submodular Transformations](https://mlsanity.com/paper/1708.08781/) (2017)
- [Axiomatic Construction of Hierarchical Clustering in Asymmetric Networks](https://mlsanity.com/paper/1301.7724/) (2013)
- [Hierarchical Clustering of Asymmetric Networks](https://mlsanity.com/paper/1607.06294/) (2016)
- [Co-clustering Vertices and Hyperedges via Spectral Hypergraph Partitioning](https://mlsanity.com/paper/2102.10169/) (2021)

## Built on by (most cited first, 1 of 1)

- [Hypergraphs as Weighted Directed Self-Looped Graphs: Spectral Properties, Clustering, Cheeger Inequality](https://mlsanity.com/paper/2411.03331/) (2024)

## Abstract (arXiv)

We propose a flexible framework for defining the 1-Laplacian of a hypergraph that incorporates edge-dependent vertex weights. These weights are able to reflect varying importance of vertices within a hyperedge, thus conferring the hypergraph model higher expressivity than homogeneous hypergraphs. We then utilize the eigenvector associated with the second smallest eigenvalue of the hypergraph 1-Laplacian to cluster the vertices. From a theoretical standpoint based on an adequately defined normalized Cheeger cut, this procedure is expected to achieve higher clustering accuracy than that based on the traditional Laplacian. Indeed, we confirm that this is the case using real-world datasets to demonstrate the effectiveness of the proposed spectral clustering approach. Moreover, we show that for a special case within our framework, the corresponding hypergraph 1-Laplacian is equivalent to the 1-Laplacian of a related graph, whose eigenvectors can be computed more efficiently, facilitating the adoption on larger datasets.