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.

翻译：本工作中，我们严谨地研究了图神经网络分数阶微分方程（FDE）模型的鲁棒性。该框架通过引入时间分数阶Caputo导数，突破了传统图神经网络（整数阶）常微分方程（ODE）模型的局限。借助分数阶微积分，我们的模型能够在特征更新过程中考虑长期记忆，这与传统图神经网络ODE模型的无记忆马尔可夫更新方式不同。在无攻击或扰动环境下，图神经网络FDE模型相比图神经网络ODE模型已展现出优越性。尽管现有文献已证实传统图神经网络ODE模型在对抗攻击下具备一定程度的稳定性与弹性，但图神经网络FDE模型在对抗条件下的鲁棒性仍鲜有研究。本文对图神经网络FDE模型的鲁棒性进行了系统评估。我们建立了理论框架，阐述了图神经网络FDE模型的鲁棒特性，并指出相较于整数阶模型，该模型在面对输入和图拓扑扰动时能维持更严格的输出扰动界限。实验评估进一步验证了图神经网络FDE模型增强的鲁棒性，凸显了其在对抗鲁棒应用中的潜力。