# Implicit Graph Neural Networks

arXiv: https://arxiv.org/abs/2009.06211 · HTML: https://mlsanity.com/paper/2009.06211/

## Key facts

- Cited by 100 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Sep 2020.

## Built on by (most cited first, 30 of 100)

- [Long Range Graph Benchmark](https://mlsanity.com/paper/2206.08164/) (2022)
- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [NodeFormer: A Scalable Graph Structure Learning Transformer for Node Classification](https://mlsanity.com/paper/2306.08385/) (2023)
- [Beltrami Flow and Neural Diffusion on Graphs](https://mlsanity.com/paper/2110.09443/) (2021)
- [On Training Implicit Models](https://mlsanity.com/paper/2111.05177/) (2021)
- [A Survey on Graph Representation Learning Methods](https://mlsanity.com/paper/2204.01855/) (2022)
- [Deep Equilibrium Optical Flow Estimation](https://mlsanity.com/paper/2204.08442/) (2022)
- [Graph Neural Networks Inspired by Classical Iterative Algorithms](https://mlsanity.com/paper/2103.06064/) (2021)
- [Equivariant Hypergraph Diffusion Neural Operators](https://mlsanity.com/paper/2207.06680/) (2022)
- [Nonsmooth Implicit Differentiation for Machine Learning and Optimization](https://mlsanity.com/paper/2106.04350/) (2021)
- [Deep Equilibrium Approaches to Diffusion Models](https://mlsanity.com/paper/2210.12867/) (2022)
- [Path Independent Equilibrium Models Can Better Exploit Test-Time Computation](https://mlsanity.com/paper/2211.09961/) (2022)
- [Transformers Generalize DeepSets and Can be Extended to Graphs and Hypergraphs](https://mlsanity.com/paper/2110.14416/) (2021)
- [EIGNN: Efficient Infinite-Depth Graph Neural Networks](https://mlsanity.com/paper/2202.10720/) (2022)
- [Linear-Time Graph Neural Networks for Scalable Recommendations](https://mlsanity.com/paper/2402.13973/) (2024)
- [Optimization-Induced Graph Implicit Nonlinear Diffusion](https://mlsanity.com/paper/2206.14418/) (2022)
- [One-step differentiation of iterative algorithms](https://mlsanity.com/paper/2305.13768/) (2023)
- [G-Refer: Graph Retrieval-Augmented Large Language Model for Explainable Recommendation](https://mlsanity.com/paper/2502.12586/) (2025)
- [LMC: Fast Training of GNNs via Subgraph Sampling with Provable Convergence](https://mlsanity.com/paper/2302.00924/) (2023)
- [Reversible and irreversible bracket-based dynamics for deep graph neural networks](https://mlsanity.com/paper/2305.15616/) (2023)
- [From Continuous Dynamics to Graph Neural Networks: Neural Diffusion and Beyond](https://mlsanity.com/paper/2310.10121/) (2023)
- [Alternating Differentiation for Optimization Layers](https://mlsanity.com/paper/2210.01802/) (2022)
- [TorchDEQ: A Library for Deep Equilibrium Models](https://mlsanity.com/paper/2310.18605/) (2023)
- [VACA: Design of Variational Graph Autoencoders for Interventional and Counterfactual Queries](https://mlsanity.com/paper/2110.14690/) (2021)
- [MGNNI: Multiscale Graph Neural Networks with Implicit Layers](https://mlsanity.com/paper/2210.08353/) (2022)
- [Equilibrium Reasoners: Learning Attractors Enables Scalable Reasoning](https://mlsanity.com/paper/2605.21488/) (2026)
- [Multiset-Equivariant Set Prediction with Approximate Implicit Differentiation](https://mlsanity.com/paper/2111.12193/) (2021)
- [Deep Equilibrium Diffusion Restoration with Parallel Sampling](https://mlsanity.com/paper/2311.11600/) (2023)
- [Simultaneous Missing Value Imputation and Structure Learning with Groups](https://mlsanity.com/paper/2110.08223/) (2021)
- [LazyGNN: Large-Scale Graph Neural Networks via Lazy Propagation](https://mlsanity.com/paper/2302.01503/) (2023)

## Abstract (arXiv)

Graph Neural Networks (GNNs) are widely used deep learning models that learn meaningful representations from graph-structured data. Due to the finite nature of the underlying recurrent structure, current GNN methods may struggle to capture long-range dependencies in underlying graphs. To overcome this difficulty, we propose a graph learning framework, called Implicit Graph Neural Networks (IGNN), where predictions are based on the solution of a fixed-point equilibrium equation involving implicitly defined "state" vectors. We use the Perron-Frobenius theory to derive sufficient conditions that ensure well-posedness of the framework. Leveraging implicit differentiation, we derive a tractable projected gradient descent method to train the framework. Experiments on a comprehensive range of tasks show that IGNNs consistently capture long-range dependencies and outperform the state-of-the-art GNN models.