# Three hypergraph eigenvector centralities

arXiv: https://arxiv.org/abs/1807.09644 · HTML: https://mlsanity.com/paper/1807.09644/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jul 2018.

## Built on by (most cited first, 11 of 11)

- [You are AllSet: A Multiset Function Framework for Hypergraph Neural Networks](https://mlsanity.com/paper/2106.13264/) (2021)
- [Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond](https://mlsanity.com/paper/2101.05510/) (2021)
- [Inference of hyperedges and overlapping communities in hypergraphs](https://mlsanity.com/paper/2204.05646/) (2022)
- [Adaptive Neural Message Passing for Inductive Learning on Hypergraphs](https://mlsanity.com/paper/2109.10683/) (2021)
- [Gaussian Processes on Hypergraphs](https://mlsanity.com/paper/2106.01982/) (2021)
- [Core-periphery Models for Hypergraphs](https://mlsanity.com/paper/2206.00783/) (2022)
- [Critical Nodes Identification in Complex Networks: A Survey](https://mlsanity.com/paper/2507.06164/) (2025)
- [Hypergraph Dissimilarity Measures](https://mlsanity.com/paper/2106.08206/) (2021)
- [Scalable tensor methods for nonuniform hypergraphs](https://mlsanity.com/paper/2306.17825/) (2023)
- [Community and hyperedge inference in multiple hypergraphs](https://mlsanity.com/paper/2505.04967/) (2025)
- [Identifying Group Anchors in Real-World Group Interactions Under Label Scarcity](https://mlsanity.com/paper/2509.20762/) (2025)

## Abstract (arXiv)

Eigenvector centrality is a standard network analysis tool for determining the importance of (or ranking of) entities in a connected system that is represented by a graph. However, many complex systems and datasets have natural multi-way interactions that are more faithfully modeled by a hypergraph. Here we extend the notion of graph eigenvector centrality to uniform hypergraphs. Traditional graph eigenvector centralities are given by a positive eigenvector of the adjacency matrix, which is guaranteed to exist by the Perron-Frobenius theorem under some mild conditions. The natural representation of a hypergraph is a hypermatrix (colloquially, a tensor). Using recently established Perron-Frobenius theory for tensors, we develop three tensor eigenvectors centralities for hypergraphs, each with different interpretations. We show that these centralities can reveal different information on real-world data by analyzing hypergraphs constructed from n-gram frequencies, co-tagging on stack exchange, and drug combinations observed in patient emergency room visits.