# Implicit Hypergraph Neural Networks: A Stable Framework for Higher-Order Relational Learning with Provable Guarantees

arXiv: https://arxiv.org/abs/2508.09427 · HTML: https://mlsanity.com/paper/2508.09427/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 53 of its 85 references that are in the map.
- First posted on arXiv 13 Aug 2025.

## Builds on (most cited first, 30 of 53)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [DeepWalk: Online Learning of Social Representations](https://mlsanity.com/paper/1403.6652/) (2014)
- [Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning](https://mlsanity.com/paper/1801.07606/) (2018)
- [Revisiting Semi-Supervised Learning with Graph Embeddings](https://mlsanity.com/paper/1603.08861/) (2016)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Graph Neural Networks in Recommender Systems: A Survey](https://mlsanity.com/paper/2011.02260/) (2020)
- [Hypergraph Neural Networks](https://mlsanity.com/paper/1809.09401/) (2018)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Improving Graph Neural Network Expressivity via Subgraph Isomorphism Counting](https://mlsanity.com/paper/2006.09252/) (2020)
- [DeeperGCN: All You Need to Train Deeper GCNs](https://mlsanity.com/paper/2006.07739/) (2020)
- [A Survey on Oversmoothing in Graph Neural Networks](https://mlsanity.com/paper/2303.10993/) (2023)
- [Hypergraph Convolution and Hypergraph Attention](https://mlsanity.com/paper/1901.08150/) (2019)
- [Graph Neural Ordinary Differential Equations](https://mlsanity.com/paper/1911.07532/) (2019)
- [A vector-contraction inequality for Rademacher complexities](https://mlsanity.com/paper/1605.00251/) (2016)
- [UniGNN: a Unified Framework for Graph and Hypergraph Neural Networks](https://mlsanity.com/paper/2105.00956/) (2021)
- [Continuous Graph Neural Networks](https://mlsanity.com/paper/1912.00967/) (2019)
- [HNHN: Hypergraph Networks with Hyperedge Neurons](https://mlsanity.com/paper/2006.12278/) (2020)
- [Implicit Graph Neural Networks](https://mlsanity.com/paper/2009.06211/) (2020)
- [Be More with Less: Hypergraph Attention Networks for Inductive Text Classification](https://mlsanity.com/paper/2011.00387/) (2020)
- [Multivariate Time Series Forecasting with Dynamic Graph Neural ODEs](https://mlsanity.com/paper/2202.08408/) (2022)
- [A Survey of Graph Neural Networks for Social Recommender Systems](https://mlsanity.com/paper/2212.04481/) (2022)
- [The Descriptive Complexity of Graph Neural Networks](https://mlsanity.com/paper/2303.04613/) (2023)
- [From Continuous Dynamics to Graph Neural Networks: Neural Diffusion and Beyond](https://mlsanity.com/paper/2310.10121/) (2023)
- [Circuit Complexity Bounds for RoPE-based Transformer Architecture](https://mlsanity.com/paper/2411.07602/) (2024)

## Built on by (most cited first, 3 of 3)

- [Adaptive Graph Learning with Transformer for Multi-Reservoir Inflow Prediction](https://mlsanity.com/paper/2511.07649/) (2025)
- [The WidthWall: A Strict Expressivity Hierarchy for Hypergraph Neural Networks](https://mlsanity.com/paper/2605.13690/) (2026)
- [From Diffusion to Reaction-Diffusion: A Dynamical-Systems View of Oversmoothing in Hypergraph Neural Networks](https://mlsanity.com/paper/2607.15773/) (2026)

## Abstract (arXiv)

Many real-world interactions are group-based rather than pairwise such as papers with multiple co-authors and users jointly engaging with items. Hypergraph neural networks have shown great promise at modeling higher-order relations, but their reliance on a fixed number of explicit message-passing layers limits long-range dependency capture and can destabilize training as depth grows. In this work, we introduce Implicit Hypergraph Neural Networks (IHGNN), which bring the implicit equilibrium formulation to hypergraphs: instead of stacking layers, IHGNN computes representations as the solution to a nonlinear fixed-point equation, enabling stable and efficient global propagation across hyperedges without deep architectures. We develop a well-posed training scheme with provable convergence, analyze the oversmoothing conditions and expressivity of the model, and derive a transductive generalization bound on hypergraphs. We further present an implicit-gradient training procedure coupled with a projection-based stabilization strategy. Extensive experiments on citation benchmarks show that IHGNN consistently outperforms strong traditional graph/hypergraph neural network baselines in both accuracy and robustness. Empirically, IHGNN is resilient to random initialization and hyperparameter variation, highlighting its strong generalization and practical value for higher-order relational learning.