# Coupling Graph Neural Networks with Fractional Order Continuous Dynamics: A Robustness Study

arXiv: https://arxiv.org/abs/2401.04331 · HTML: https://mlsanity.com/paper/2401.04331/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 79 references that are in the map.
- First posted on arXiv 9 Jan 2024.

## Builds on (most cited first, 30 of 40)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [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)
- [Adversarial Attacks on Neural Networks for Graph Data](https://mlsanity.com/paper/1805.07984/) (2018)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [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 Attack on Graph Structured Data](https://mlsanity.com/paper/1806.02371/) (2018)
- [Adversarial Attacks on Graph Neural Networks via Meta Learning](https://mlsanity.com/paper/1902.08412/) (2019)
- [GNNGuard: Defending Graph Neural Networks against Adversarial Attacks](https://mlsanity.com/paper/2006.08149/) (2020)
- [Graph Neural Ordinary Differential Equations](https://mlsanity.com/paper/1911.07532/) (2019)
- [A Semi-supervised Graph Attentive Network for Financial Fraud Detection](https://mlsanity.com/paper/2003.01171/) (2020)
- [A Comprehensive Survey on Trustworthy Graph Neural Networks: Privacy, Robustness, Fairness, and Explainability](https://mlsanity.com/paper/2204.08570/) (2022)
- [Continuous Graph Neural Networks](https://mlsanity.com/paper/1912.00967/) (2019)
- [Topology Adaptive Graph Convolutional Networks](https://mlsanity.com/paper/1710.10370/) (2017)
- [Robustness of Graph Neural Networks at Scale](https://mlsanity.com/paper/2110.14038/) (2021)
- [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)
- [DeepRobust: A PyTorch Library for Adversarial Attacks and Defenses](https://mlsanity.com/paper/2005.06149/) (2020)
- [Hiding Individuals and Communities in a Social Network](https://mlsanity.com/paper/1608.00375/) (2016)
- [Towards More Practical Adversarial Attacks on Graph Neural Networks](https://mlsanity.com/paper/2006.05057/) (2020)
- [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)
- [Understanding and Improving Graph Injection Attack by Promoting Unnoticeability](https://mlsanity.com/paper/2202.08057/) (2022)
- [GREAD: Graph Neural Reaction-Diffusion Networks](https://mlsanity.com/paper/2211.14208/) (2022)

## Built on by (most cited first, 9 of 9)

- [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)
- [Distributed-Order Fractional Graph Operating Network](https://mlsanity.com/paper/2411.05274/) (2024)
- [Neural Variable-Order Fractional Differential Equation Networks](https://mlsanity.com/paper/2503.16207/) (2025)
- [Efficient Training of Neural Fractional-Order Differential Equation via Adjoint Backpropagation](https://mlsanity.com/paper/2503.16666/) (2025)
- [Learning Aligned Stability in Neural ODEs Reconciling Accuracy with Robustness](https://mlsanity.com/paper/2509.21879/) (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)
- [LongSpike: Fractional Order Spiking State Space Models for Efficient Long Sequence Learning](https://mlsanity.com/paper/2606.12895/) (2026)

## Abstract (arXiv)

In this work, we rigorously investigate the robustness of graph neural fractional-order differential equation (FDE) models. This framework extends beyond traditional graph neural (integer-order) ordinary differential equation (ODE) models by implementing the time-fractional Caputo derivative. Utilizing fractional calculus allows our model to consider long-term memory during the feature updating process, diverging from the memoryless Markovian updates seen in traditional graph neural ODE models. The superiority of graph neural FDE models over graph neural ODE models has been established in environments free from attacks or perturbations. While traditional graph neural ODE models have been verified to possess a degree of stability and resilience in the presence of adversarial attacks in existing literature, the robustness of graph neural FDE models, especially under adversarial conditions, remains largely unexplored. This paper undertakes a detailed assessment of the robustness of graph neural FDE models. We establish a theoretical foundation outlining the robustness characteristics of graph neural FDE models, highlighting that they maintain more stringent output perturbation bounds in the face of input and graph topology disturbances, compared to their integer-order counterparts. Our empirical evaluations further confirm the enhanced robustness of graph neural FDE models, highlighting their potential in adversarially robust applications.