# From Hypergraph Energy Functions to Hypergraph Neural Networks

arXiv: https://arxiv.org/abs/2306.09623 · HTML: https://mlsanity.com/paper/2306.09623/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 52 references that are in the map.
- First posted on arXiv 16 Jun 2023.

## Builds on (most cited first, 28 of 28)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Modeling Relational Data with Graph Convolutional Networks](https://mlsanity.com/paper/1703.06103/) (2017)
- [Heterogeneous Graph Attention Network](https://mlsanity.com/paper/1903.07293/) (2019)
- [Multi-view Convolutional Neural Networks for 3D Shape Recognition](https://mlsanity.com/paper/1505.00880/) (2015)
- [Hypergraph Neural Networks](https://mlsanity.com/paper/1809.09401/) (2018)
- [Hypergraph Convolution and Hypergraph Attention](https://mlsanity.com/paper/1901.08150/) (2019)
- [Interpreting and Unifying Graph Neural Networks with An Optimization Framework](https://mlsanity.com/paper/2101.11859/) (2021)
- [UniGNN: a Unified Framework for Graph and Hypergraph Neural Networks](https://mlsanity.com/paper/2105.00956/) (2021)
- [You are AllSet: A Multiset Function Framework for Hypergraph Neural Networks](https://mlsanity.com/paper/2106.13264/) (2021)
- [Higher-order organization of complex networks](https://mlsanity.com/paper/1612.08447/) (2016)
- [A Unified View on Graph Neural Networks as Graph Signal Denoising](https://mlsanity.com/paper/2010.01777/) (2020)
- [HNHN: Hypergraph Networks with Hyperedge Neurons](https://mlsanity.com/paper/2006.12278/) (2020)
- [Be More with Less: Hypergraph Attention Networks for Inductive Text Classification](https://mlsanity.com/paper/2011.00387/) (2020)
- [Elastic Graph Neural Networks](https://mlsanity.com/paper/2107.06996/) (2021)
- [Graph Neural Networks Inspired by Classical Iterative Algorithms](https://mlsanity.com/paper/2103.06064/) (2021)
- [HyperSAGE: Generalizing Inductive Representation Learning on Hypergraphs](https://mlsanity.com/paper/2010.04558/) (2020)
- [Generative hypergraph clustering: from blockmodels to modularity](https://mlsanity.com/paper/2101.09611/) (2021)
- [Equivariant Hypergraph Diffusion Neural Operators](https://mlsanity.com/paper/2207.06680/) (2022)
- [Inhomogeneous Hypergraph Clustering with Applications](https://mlsanity.com/paper/1709.01249/) (2017)
- [Clustering in graphs and hypergraphs with categorical edge labels](https://mlsanity.com/paper/1910.09943/) (2019)
- [Revisiting Graph Convolutional Network on Semi-Supervised Node Classification from an Optimization Perspective](https://mlsanity.com/paper/2009.11469/) (2020)
- [Hypergraph Convolutional Networks via Equivalency between Hypergraphs and Undirected Graphs](https://mlsanity.com/paper/2203.16939/) (2022)
- [The Spacey Random Walk: a Stochastic Process for Higher-order Data](https://mlsanity.com/paper/1602.02102/) (2016)
- [Learning over Families of Sets -- Hypergraph Representation Learning for Higher Order Tasks](https://mlsanity.com/paper/2101.07773/) (2021)
- [HEAT: Hyperedge Attention Networks](https://mlsanity.com/paper/2201.12113/) (2022)
- [Message Passing Neural Networks for Hypergraphs](https://mlsanity.com/paper/2203.16995/) (2022)
- [Descent Steps of a Relation-Aware Energy Produce Heterogeneous Graph Neural Networks](https://mlsanity.com/paper/2206.11081/) (2022)
- [A Hypergraph Neural Network Framework for Learning Hyperedge-Dependent Node Embeddings](https://mlsanity.com/paper/2212.14077/) (2022)

## Built on by (most cited first, 18 of 18)

- [A Survey on Hypergraph Neural Networks: An In-Depth and Step-By-Step Guide](https://mlsanity.com/paper/2404.01039/) (2024)
- [HypeBoy: Generative Self-Supervised Representation Learning on Hypergraphs](https://mlsanity.com/paper/2404.00638/) (2024)
- [4DBInfer: A 4D Benchmarking Toolbox for Graph-Centric Predictive Modeling on Relational DBs](https://mlsanity.com/paper/2404.18209/) (2024)
- [Training-Free Message Passing for Learning on Hypergraphs](https://mlsanity.com/paper/2402.05569/) (2024)
- [DHG-Bench: A Comprehensive Benchmark for Deep Hypergraph Learning](https://mlsanity.com/paper/2508.12244/) (2025)
- [Ada-HGNN: Adaptive Sampling for Scalable Hypergraph Neural Networks](https://mlsanity.com/paper/2405.13372/) (2024)
- [Hypergraph Neural Diffusion: A PDE-Inspired Framework for Hypergraph Message Passing](https://mlsanity.com/paper/2604.10955/) (2026)
- [Modeling Edge-Specific Node Features through Co-Representation Neural Hypergraph Diffusion](https://mlsanity.com/paper/2405.14286/) (2024)
- [Heterophily-Agnostic Hypergraph Neural Networks with Riemannian Local Exchanger](https://mlsanity.com/paper/2603.00599/) (2026)
- [DLGNet: Hyperedge Classification through Directed Line Graphs for Chemical Reactions](https://mlsanity.com/paper/2410.06969/) (2024)
- [Implicit Hypergraph Neural Network](https://mlsanity.com/paper/2508.14101/) (2025)
- [Disentangling Hyperedges through the Lens of Category Theory](https://mlsanity.com/paper/2510.16289/) (2025)
- [Anchor-guided Hypergraph Condensation with Dual-level Discrimination](https://mlsanity.com/paper/2605.10001/) (2026)
- [Hypergraph Pattern Machine: Compositional Tokenization for Higher-Order Interactions](https://mlsanity.com/paper/2605.16527/) (2026)
- [Hypergraph Neural Stochastic Diffusion: An SDE Framework for Uncertainty Estimation](https://mlsanity.com/paper/2607.07330/) (2026)
- [From Diffusion to Reaction-Diffusion: A Dynamical-Systems View of Oversmoothing in Hypergraph Neural Networks](https://mlsanity.com/paper/2607.15773/) (2026)
- [α-Graph: Attention-Infused Normalizing Flow Approach to Tractable Graph Modeling](https://mlsanity.com/paper/2609.07961/) (2026)
- [PANDA: Prior-guided Attentional Dual-path Architecture](https://mlsanity.com/paper/2610.13635/) (2026)

## Abstract (arXiv)

Hypergraphs are a powerful abstraction for representing higher-order interactions between entities of interest. To exploit these relationships in making downstream predictions, a variety of hypergraph neural network architectures have recently been proposed, in large part building upon precursors from the more traditional graph neural network (GNN) literature. Somewhat differently, in this paper we begin by presenting an expressive family of parameterized, hypergraph-regularized energy functions. We then demonstrate how minimizers of these energies effectively serve as node embeddings that, when paired with a parameterized classifier, can be trained end-to-end via a supervised bilevel optimization process. Later, we draw parallels between the implicit architecture of the predictive models emerging from the proposed bilevel hypergraph optimization, and existing GNN architectures in common use. Empirically, we demonstrate state-of-the-art results on various hypergraph node classification benchmarks. Code is available at https://github.com/yxzwang/PhenomNN.