# EIGNN: Efficient Infinite-Depth Graph Neural Networks

arXiv: https://arxiv.org/abs/2202.10720 · HTML: https://mlsanity.com/paper/2202.10720/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 40 references that are in the map.
- First posted on arXiv 22 Feb 2022.

## Builds on (most cited first, 27 of 27)

- [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)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [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)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning](https://mlsanity.com/paper/1801.07606/) (2018)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (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)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [Gated Graph Sequence Neural Networks](https://mlsanity.com/paper/1511.05493/) (2015)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Multi-scale Attributed Node Embedding](https://mlsanity.com/paper/1909.13021/) (2019)
- [Adversarial Attacks on Neural Networks for Graph Data](https://mlsanity.com/paper/1805.07984/) (2018)
- [Deep Learning on Graphs: A Survey](https://mlsanity.com/paper/1812.04202/) (2018)
- [PairNorm: Tackling Oversmoothing in GNNs](https://mlsanity.com/paper/1909.12223/) (2019)
- [Graph Information Bottleneck](https://mlsanity.com/paper/2010.12811/) (2020)
- [Multiscale Deep Equilibrium Models](https://mlsanity.com/paper/2006.08656/) (2020)
- [Implicit Deep Learning](https://mlsanity.com/paper/1908.06315/) (2019)
- [Implicit Graph Neural Networks](https://mlsanity.com/paper/2009.06211/) (2020)
- [Scaling Up Graph Neural Networks Via Graph Coarsening](https://mlsanity.com/paper/2106.05150/) (2021)
- [Graph Coarsening with Neural Networks](https://mlsanity.com/paper/2102.01350/) (2021)
- [On the Theory of Implicit Deep Learning: Global Convergence with Implicit Layers](https://mlsanity.com/paper/2102.07346/) (2021)

## Built on by (most cited first, 30 of 32)

- [Long Range Graph Benchmark](https://mlsanity.com/paper/2206.08164/) (2022)
- [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)
- [Rank Diminishing in Deep Neural Networks](https://mlsanity.com/paper/2206.06072/) (2022)
- [Equivariant Hypergraph Diffusion Neural Operators](https://mlsanity.com/paper/2207.06680/) (2022)
- [On Vanishing Gradients, Over-Smoothing, and Over-Squashing in GNNs: Bridging Recurrent and Graph Learning](https://mlsanity.com/paper/2502.10818/) (2025)
- [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)
- [Optimization-Induced Graph Implicit Nonlinear Diffusion](https://mlsanity.com/paper/2206.14418/) (2022)
- [Adaptive Message Passing: A General Framework to Mitigate Oversmoothing, Oversquashing, and Underreaching](https://mlsanity.com/paper/2312.16560/) (2023)
- [TorchDEQ: A Library for Deep Equilibrium Models](https://mlsanity.com/paper/2310.18605/) (2023)
- [MGNNI: Multiscale Graph Neural Networks with Implicit Layers](https://mlsanity.com/paper/2210.08353/) (2022)
- [Single-Pass Contrastive Learning Can Work for Both Homophilic and Heterophilic Graph](https://mlsanity.com/paper/2211.10890/) (2022)
- [From Graphs to Qubits: A Critical Review of Quantum Graph Neural Networks](https://mlsanity.com/paper/2408.06524/) (2024)
- [Bridging Trustworthiness and Open-World Learning: An Exploratory Neural Approach for Enhancing Interpretability, Generalization, and Robustness](https://mlsanity.com/paper/2308.03666/) (2023)
- [OpenViewer: Openness-Aware Multi-View Learning](https://mlsanity.com/paper/2412.12596/) (2024)
- [Efficient and Effective Implicit Dynamic Graph Neural Network](https://mlsanity.com/paper/2406.17894/) (2024)
- [IGNN-Solver: A Graph Neural Solver for Implicit Graph Neural Networks](https://mlsanity.com/paper/2410.08524/) (2024)
- [ADMP-GNN: Adaptive Depth Message Passing GNN](https://mlsanity.com/paper/2509.01170/) (2025)
- [Implicit Graph Neural Diffusion Networks: Convergence, Generalization, and Over-Smoothing](https://mlsanity.com/paper/2308.03306/) (2023)
- [Graph Neural Networks Gone Hogwild](https://mlsanity.com/paper/2407.00494/) (2024)
- [Consistency Deep Equilibrium Models](https://mlsanity.com/paper/2602.03024/) (2026)
- [Unifying Label-inputted Graph Neural Networks with Deep Equilibrium Models](https://mlsanity.com/paper/2211.10629/) (2022)
- [Towards a GML-Enabled Knowledge Graph Platform](https://mlsanity.com/paper/2303.02166/) (2023)
- [Revisiting Generalized p-Laplacian Regularized Framelet GCNs: Convergence, Energy Dynamic and Training with Non-Linear Diffusion](https://mlsanity.com/paper/2305.15639/) (2023)
- [Hybrid Quantum Graph Neural Network for Molecular Property Prediction](https://mlsanity.com/paper/2405.05205/) (2024)
- [Lipschitz Multiscale Deep Equilibrium Models: A Theoretically Guaranteed and Accelerated Approach](https://mlsanity.com/paper/2602.03297/) (2026)
- [Generalized Laplacian Regularized Framelet Graph Neural Networks](https://mlsanity.com/paper/2210.15092/) (2022)
- [Provably Convergent Subgraph-wise Sampling for Fast GNN Training](https://mlsanity.com/paper/2303.11081/) (2023)
- [Implicit Hypergraph Neural Network](https://mlsanity.com/paper/2508.14101/) (2025)

## Abstract (arXiv)

Graph neural networks (GNNs) are widely used for modelling graph-structured data in numerous applications. However, with their inherently finite aggregation layers, existing GNN models may not be able to effectively capture long-range dependencies in the underlying graphs. Motivated by this limitation, we propose a GNN model with infinite depth, which we call Efficient Infinite-Depth Graph Neural Networks (EIGNN), to efficiently capture very long-range dependencies. We theoretically derive a closed-form solution of EIGNN which makes training an infinite-depth GNN model tractable. We then further show that we can achieve more efficient computation for training EIGNN by using eigendecomposition. The empirical results of comprehensive experiments on synthetic and real-world datasets show that EIGNN has a better ability to capture long-range dependencies than recent baselines, and consistently achieves state-of-the-art performance. Furthermore, we show that our model is also more robust against both noise and adversarial perturbations on node features.