# Unifying Label-inputted Graph Neural Networks with Deep Equilibrium Models

arXiv: https://arxiv.org/abs/2211.10629 · HTML: https://mlsanity.com/paper/2211.10629/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 40 references that are in the map.
- First posted on arXiv 19 Nov 2022.

## Builds on (most cited first, 30 of 31)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [Pitfalls of Graph Neural Network Evaluation](https://mlsanity.com/paper/1811.05868/) (2018)
- [Representation Learning on Graphs with Jumping Knowledge Networks](https://mlsanity.com/paper/1806.03536/) (2018)
- [Simple and Deep Graph Convolutional Networks](https://mlsanity.com/paper/2007.02133/) (2020)
- [Graph WaveNet for Deep Spatial-Temporal Graph Modeling](https://mlsanity.com/paper/1906.00121/) (2019)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [Image-based Recommendations on Styles and Substitutes](https://mlsanity.com/paper/1506.04757/) (2015)
- [GraphSAINT: Graph Sampling Based Inductive Learning Method](https://mlsanity.com/paper/1907.04931/) (2019)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [FastGCN: Fast Learning with Graph Convolutional Networks via Importance Sampling](https://mlsanity.com/paper/1801.10247/) (2018)
- [Multi-scale Attributed Node Embedding](https://mlsanity.com/paper/1909.13021/) (2019)
- [DeepGCNs: Can GCNs Go as Deep as CNNs?](https://mlsanity.com/paper/1904.03751/) (2019)
- [Masked Label Prediction: Unified Message Passing Model for Semi-Supervised Classification](https://mlsanity.com/paper/2009.03509/) (2020)
- [Combining Label Propagation and Simple Models Out-performs Graph Neural Networks](https://mlsanity.com/paper/2010.13993/) (2020)
- [On the Properties of the Softmax Function with Application in Game Theory and Reinforcement Learning](https://mlsanity.com/paper/1704.00805/) (2017)
- [Monotone operator equilibrium networks](https://mlsanity.com/paper/2006.08591/) (2020)
- [Implicit Graph Neural Networks](https://mlsanity.com/paper/2009.06211/) (2020)
- [Unifying Graph Convolutional Neural Networks and Label Propagation](https://mlsanity.com/paper/2002.06755/) (2020)
- [EIGNN: Efficient Infinite-Depth Graph Neural Networks](https://mlsanity.com/paper/2202.10720/) (2022)
- [On the Global Convergence of Training Deep Linear ResNets](https://mlsanity.com/paper/2003.01094/) (2020)
- [MGNNI: Multiscale Graph Neural Networks with Implicit Layers](https://mlsanity.com/paper/2210.08353/) (2022)
- [Prototype Propagation Networks (PPN) for Weakly-supervised Few-shot Learning on Category Graph](https://mlsanity.com/paper/1905.04042/) (2019)

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

- [Graph Entropy Minimization for Semi-supervised Node Classification](https://mlsanity.com/paper/2305.19502/) (2023)

## Abstract (arXiv)

The success of Graph Neural Networks (GNN) in learning on non-Euclidean data arouses many subtopics, such as Label-inputted GNN (LGNN) and Implicit GNN (IGNN). LGNN, explicitly inputting supervising information (a.k.a. labels) in GNN, integrates label propagation to achieve superior performance, but with the dilemma between its propagating distance and adaptiveness. IGNN, outputting an equilibrium point by iterating its network infinite times, exploits information in the entire graph to capture long-range dependencies, but with its network constrained to guarantee the existence of the equilibrium. This work unifies the two subdomains by interpreting LGNN in the theory of IGNN and reducing prevailing LGNNs to the form of IGNN. The unification facilitates the exchange between the two subdomains and inspires more studies. Specifically, implicit differentiation of IGNN is introduced to LGNN to differentiate its infinite-range label propagation with constant memory, making the propagation both distant and adaptive. Besides, the masked label strategy of LGNN is proven able to guarantee the well-posedness of IGNN in a network-agnostic manner, granting its network more complex and thus more expressive. Combining the advantages of LGNN and IGNN, Label-inputted Implicit GNN (LI-GNN) is proposed. It can be widely applied to any specific GNN to boost its performance. Node classification experiments on two synthesized and six real-world datasets demonstrate its effectiveness. Code is available at https://github.com/cf020031308/LI-GNN