# On the Robustness of Graph Neural Diffusion to Topology Perturbations

arXiv: https://arxiv.org/abs/2209.07754 · HTML: https://mlsanity.com/paper/2209.07754/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 60 references that are in the map.
- First posted on arXiv 16 Sep 2022.

## Builds on (most cited first, 30 of 33)

- [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)
- [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)
- [Variational Graph Auto-Encoders](https://mlsanity.com/paper/1611.07308/) (2016)
- [Graph Convolutional Neural Networks for Web-Scale Recommender Systems](https://mlsanity.com/paper/1806.01973/) (2018)
- [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)
- [Image-based Recommendations on Styles and Substitutes](https://mlsanity.com/paper/1506.04757/) (2015)
- [Understanding over-squashing and bottlenecks on graphs via curvature](https://mlsanity.com/paper/2111.14522/) (2021)
- [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)
- [Graph Structure Learning for Robust Graph Neural Networks](https://mlsanity.com/paper/2005.10203/) (2020)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Adversarial Attacks on Graph Neural Networks via Meta Learning](https://mlsanity.com/paper/1902.08412/) (2019)
- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [Graph Random Neural Network for Semi-Supervised Learning on Graphs](https://mlsanity.com/paper/2005.11079/) (2020)
- [GNNGuard: Defending Graph Neural Networks against Adversarial Attacks](https://mlsanity.com/paper/2006.08149/) (2020)
- [Topology Adaptive Graph Convolutional Networks](https://mlsanity.com/paper/1710.10370/) (2017)
- [Stable Neural ODE with Lyapunov-Stable Equilibrium Points for Defending Against Adversarial Attacks](https://mlsanity.com/paper/2110.12976/) (2021)
- [Beltrami Flow and Neural Diffusion on Graphs](https://mlsanity.com/paper/2110.09443/) (2021)
- [On Robustness of Neural Ordinary Differential Equations](https://mlsanity.com/paper/1910.05513/) (2019)
- [Hiding Individuals and Communities in a Social Network](https://mlsanity.com/paper/1608.00375/) (2016)
- [TDGIA:Effective Injection Attacks on Graph Neural Networks](https://mlsanity.com/paper/2106.06663/) (2021)
- [Fast Gradient Attack on Network Embedding](https://mlsanity.com/paper/1809.02797/) (2018)
- [Graph Geometry Interaction Learning](https://mlsanity.com/paper/2010.12135/) (2020)
- [Graph Embedding on Biomedical Networks: Methods, Applications, and Evaluations](https://mlsanity.com/paper/1906.05017/) (2019)

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

- [Graph Neural Convection-Diffusion with Heterophily](https://mlsanity.com/paper/2305.16780/) (2023)
- [From Continuous Dynamics to Graph Neural Networks: Neural Diffusion and Beyond](https://mlsanity.com/paper/2310.10121/) (2023)
- [Adversarial Robustness in Graph Neural Networks: A Hamiltonian Approach](https://mlsanity.com/paper/2310.06396/) (2023)
- [RobustLoc: Robust Camera Pose Regression in Challenging Driving Environments](https://mlsanity.com/paper/2211.11238/) (2022)
- [Unleashing the Potential of Fractional Calculus in Graph Neural Networks with FROND](https://mlsanity.com/paper/2404.17099/) (2024)
- [Graph ODEs and Beyond: A Comprehensive Survey on Integrating Differential Equations with Graph Neural Networks](https://mlsanity.com/paper/2503.23167/) (2025)
- [A Generalized Neural Diffusion Framework on Graphs](https://mlsanity.com/paper/2312.08616/) (2023)
- [Node Embedding from Neural Hamiltonian Orbits in Graph Neural Networks](https://mlsanity.com/paper/2305.18965/) (2023)
- [Leveraging Label Non-Uniformity for Node Classification in Graph Neural Networks](https://mlsanity.com/paper/2305.00139/) (2023)
- [Coupling Graph Neural Networks with Fractional Order Continuous Dynamics: A Robustness Study](https://mlsanity.com/paper/2401.04331/) (2024)
- [Understanding Oversmoothing in Diffusion-Based GNNs From the Perspective of Operator Semigroup Theory](https://mlsanity.com/paper/2402.15326/) (2024)
- [Continuous Product Graph Neural Networks](https://mlsanity.com/paper/2405.18877/) (2024)
- [Distributed-Order Fractional Graph Operating Network](https://mlsanity.com/paper/2411.05274/) (2024)
- [PointDifformer: Robust Point Cloud Registration With Neural Diffusion and Transformer](https://mlsanity.com/paper/2404.14034/) (2024)
- [Neural Variable-Order Fractional Differential Equation Networks](https://mlsanity.com/paper/2503.16207/) (2025)
- [RobustMat: Neural Diffusion for Street Landmark Patch Matching under Challenging Environments](https://mlsanity.com/paper/2311.03904/) (2023)
- [SAGMAN: Stability Analysis of Graph Neural Networks on the Manifolds](https://mlsanity.com/paper/2402.08653/) (2024)
- [PosDiffNet: Positional Neural Diffusion for Point Cloud Registration in a Large Field of View with Perturbations](https://mlsanity.com/paper/2401.03167/) (2024)
- [Continuous Geometry-Aware Graph Diffusion via Hyperbolic Neural PDE](https://mlsanity.com/paper/2406.01282/) (2024)
- [PRFusion: Toward Effective and Robust Multi-Modal Place Recognition with Image and Point Cloud Fusion](https://mlsanity.com/paper/2410.04939/) (2024)
- [Continuous Simplicial Neural Networks](https://mlsanity.com/paper/2503.12919/) (2025)
- [GKNet: Graph Kalman Filtering and Model Inference via Model-based Deep Learning](https://mlsanity.com/paper/2506.22004/) (2025)
- [Resilient Graph Neural Networks: A Coupled Dynamical Systems Approach](https://mlsanity.com/paper/2311.06942/) (2023)
- [Adverseness vs. Equilibrium: Exploring Graph Adversarial Resilience through Dynamic Equilibrium](https://mlsanity.com/paper/2505.14463/) (2025)
- [Node Embedding from Hamiltonian Information Propagation in Graph Neural Networks](https://mlsanity.com/paper/2303.01030/) (2023)
- [Generalized Graphon Process: Convergence of Graph Frequencies in Stretched Cut Distance](https://mlsanity.com/paper/2309.05260/) (2023)
- [Simple Graph Contrastive Learning via Fractional-order Neural Diffusion Networks](https://mlsanity.com/paper/2504.16748/) (2025)
- [Second-Order Tensorial Partial Differential Equations on Graphs](https://mlsanity.com/paper/2509.02015/) (2025)
- [Adaptive Multi-view Graph Contrastive Learning via Fractional-order Neural Diffusion Networks](https://mlsanity.com/paper/2511.06216/) (2025)
- [Lyapunov Stable Graph Neural Flow](https://mlsanity.com/paper/2603.12557/) (2026)

## Abstract (arXiv)

Neural diffusion on graphs is a novel class of graph neural networks that has attracted increasing attention recently. The capability of graph neural partial differential equations (PDEs) in addressing common hurdles of graph neural networks (GNNs), such as the problems of over-smoothing and bottlenecks, has been investigated but not their robustness to adversarial attacks. In this work, we explore the robustness properties of graph neural PDEs. We empirically demonstrate that graph neural PDEs are intrinsically more robust against topology perturbation as compared to other GNNs. We provide insights into this phenomenon by exploiting the stability of the heat semigroup under graph topology perturbations. We discuss various graph diffusion operators and relate them to existing graph neural PDEs. Furthermore, we propose a general graph neural PDE framework based on which a new class of robust GNNs can be defined. We verify that the new model achieves comparable state-of-the-art performance on several benchmark datasets.