# Elastic Graph Neural Networks

arXiv: https://arxiv.org/abs/2107.06996 · HTML: https://mlsanity.com/paper/2107.06996/

## Key facts

- Cited by 67 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 50 references that are in the map.
- First posted on arXiv 5 Jul 2021.

## Builds on (most cited first, 25 of 25)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [Pitfalls of Graph Neural Network Evaluation](https://mlsanity.com/paper/1811.05868/) (2018)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [Adversarial Attacks on Neural Networks for Graph Data](https://mlsanity.com/paper/1805.07984/) (2018)
- [Graph Structure Learning for Robust Graph Neural Networks](https://mlsanity.com/paper/2005.10203/) (2020)
- [PairNorm: Tackling Oversmoothing in GNNs](https://mlsanity.com/paper/1909.12223/) (2019)
- [Adversarial Attacks on Graph Neural Networks via Meta Learning](https://mlsanity.com/paper/1902.08412/) (2019)
- [Revisiting Graph Neural Networks: All We Have is Low-Pass Filters](https://mlsanity.com/paper/1905.09550/) (2019)
- [Interpreting and Unifying Graph Neural Networks with An Optimization Framework](https://mlsanity.com/paper/2101.11859/) (2021)
- [A Unified View on Graph Neural Networks as Graph Signal Denoising](https://mlsanity.com/paper/2010.01777/) (2020)
- [DeepRobust: A PyTorch Library for Adversarial Attacks and Defenses](https://mlsanity.com/paper/2005.06149/) (2020)
- [Adaptive piecewise polynomial estimation via trend filtering](https://mlsanity.com/paper/1304.2986/) (2013)
- [Trend Filtering on Graphs](https://mlsanity.com/paper/1410.7690/) (2014)
- [Graph Unrolling Networks: Interpretable Neural Networks for Graph Signal Denoising](https://mlsanity.com/paper/2006.01301/) (2020)
- [Vector-Valued Graph Trend Filtering with Non-Convex Penalties](https://mlsanity.com/paper/1905.12692/) (2019)
- [Semi-supervised Learning in Network-Structured Data via Total Variation Minimization](https://mlsanity.com/paper/1901.09838/) (2019)
- [Multiclass Total Variation Clustering](https://mlsanity.com/paper/1306.1185/) (2013)
- [An Adaptive Total Variation Algorithm for Computing the Balanced Cut of a Graph](https://mlsanity.com/paper/1302.2717/) (2013)
- [New convergence analysis of a primal-dual algorithm with large stepsizes](https://mlsanity.com/paper/1711.06785/) (2017)

## Built on by (most cited first, 30 of 67)

- [Graph Condensation for Graph Neural Networks](https://mlsanity.com/paper/2110.07580/) (2021)
- [Condensing Graphs via One-Step Gradient Matching](https://mlsanity.com/paper/2206.07746/) (2022)
- [A Survey on Graph Representation Learning Methods](https://mlsanity.com/paper/2204.01855/) (2022)
- [Cluster-guided Contrastive Graph Clustering Network](https://mlsanity.com/paper/2301.01098/) (2023)
- [Graph Trend Filtering Networks for Recommendations](https://mlsanity.com/paper/2108.05552/) (2021)
- [Equivariant Hypergraph Diffusion Neural Operators](https://mlsanity.com/paper/2207.06680/) (2022)
- [Linkless Link Prediction via Relational Distillation](https://mlsanity.com/paper/2210.05801/) (2022)
- [Are Defenses for Graph Neural Networks Robust?](https://mlsanity.com/paper/2301.13694/) (2023)
- [FMP: Toward Fair Graph Message Passing against Topology Bias](https://mlsanity.com/paper/2202.04187/) (2022)
- [Reliable Representations Make A Stronger Defender: Unsupervised Structure Refinement for Robust GNN](https://mlsanity.com/paper/2207.00012/) (2022)
- [A Review of Graph Neural Networks in Epidemic Modeling](https://mlsanity.com/paper/2403.19852/) (2024)
- [Optimization-Induced Graph Implicit Nonlinear Diffusion](https://mlsanity.com/paper/2206.14418/) (2022)
- [Transformers from an Optimization Perspective](https://mlsanity.com/paper/2205.13891/) (2022)
- [A Survey of Trustworthy Graph Learning: Reliability, Explainability, and Privacy Protection](https://mlsanity.com/paper/2205.10014/) (2022)
- [Feature Overcorrelation in Deep Graph Neural Networks: A New Perspective](https://mlsanity.com/paper/2206.07743/) (2022)
- [Jointly Attacking Graph Neural Network and its Explanations](https://mlsanity.com/paper/2108.03388/) (2021)
- [From Hypergraph Energy Functions to Hypergraph Neural Networks](https://mlsanity.com/paper/2306.09623/) (2023)
- [Test-Time Training for Graph Neural Networks](https://mlsanity.com/paper/2210.08813/) (2022)
- [Defending Graph Convolutional Networks against Dynamic Graph Perturbations via Bayesian Self-supervision](https://mlsanity.com/paper/2203.03762/) (2022)
- [LazyGNN: Large-Scale Graph Neural Networks via Lazy Propagation](https://mlsanity.com/paper/2302.01503/) (2023)
- [A Unified Framework of Graph Information Bottleneck for Robustness and Membership Privacy](https://mlsanity.com/paper/2306.08604/) (2023)
- [Implicit vs Unfolded Graph Neural Networks](https://mlsanity.com/paper/2111.06592/) (2021)
- [Robust Node Classification on Graphs: Jointly from Bayesian Label Transition and Topology-based Label Propagation](https://mlsanity.com/paper/2208.09779/) (2022)
- [Chasing Fairness in Graphs: A GNN Architecture Perspective](https://mlsanity.com/paper/2312.12369/) (2023)
- [Recent Advances in Reliable Deep Graph Learning: Inherent Noise, Distribution Shift, and Adversarial Attack](https://mlsanity.com/paper/2202.07114/) (2022)
- [Bounding the Expected Robustness of Graph Neural Networks Subject to Node Feature Attacks](https://mlsanity.com/paper/2404.17947/) (2024)
- [Graph Denoising with Framelet Regularizer](https://mlsanity.com/paper/2111.03264/) (2021)
- [MGNN: Graph Neural Networks Inspired by Distance Geometry Problem](https://mlsanity.com/paper/2201.12994/) (2022)
- [Robust Graph Neural Networks via Unbiased Aggregation](https://mlsanity.com/paper/2311.14934/) (2023)
- [FocusedCleaner: Sanitizing Poisoned Graphs for Robust GNN-based Node Classification](https://mlsanity.com/paper/2210.13815/) (2022)

## Abstract (arXiv)

While many existing graph neural networks (GNNs) have been proven to perform ℓ2-based graph smoothing that enforces smoothness globally, in this work we aim to further enhance the local smoothness adaptivity of GNNs via ℓ1-based graph smoothing. As a result, we introduce a family of GNNs (Elastic GNNs) based on ℓ1 and ℓ2-based graph smoothing. In particular, we propose a novel and general message passing scheme into GNNs. This message passing algorithm is not only friendly to back-propagation training but also achieves the desired smoothing properties with a theoretical convergence guarantee. Experiments on semi-supervised learning tasks demonstrate that the proposed Elastic GNNs obtain better adaptivity on benchmark datasets and are significantly robust to graph adversarial attacks. The implementation of Elastic GNNs is available at urlhttps://github.com/lxiaorui/ElasticGNN.