We introduce the FRactional-Order graph Neural Dynamical network (FROND), a
new continuous graph neural network (GNN) framework. Unlike traditional
continuous GNNs that rely on integer-order differential equations, FROND
employs the Caputo fractional derivative to leverage the non-local properties
of fractional calculus. This approach enables the capture of long-term
dependencies in feature updates, moving beyond the Markovian update mechanisms
in conventional integer-order models and offering enhanced capabilities in
graph representation learning. We offer an interpretation of the node feature
updating process in FROND from a non-Markovian random walk perspective when the
feature updating is particularly governed by a diffusion process. We
demonstrate analytically that oversmoothing can be mitigated in this setting.
Experimentally, we validate the FROND framework by comparing the fractional
adaptations of various established integer-order continuous GNNs, demonstrating
their consistently improved performance and underscoring the framework's
potential as an effective extension to enhance traditional continuous GNNs. The
code is available at https://github.com/zknus/ICLR2024-FROND.

我们介绍了 FROND（FRactional-Order graph Neural Dynamical network），这是一个新的连续图神经网络（GNN）框架，通过使用 Caputo 分数阶导数来利用分数阶微积分的非局部特性，使得该方法能够捕捉长期依赖的特征更新，并提供了在图表示学习中超越传统整数阶模型的增强能力。我们从非马尔科夫随机游走的角度解释了 FROND 中的节点特征更新过程，特别是在特征更新受扩散过程约束时。我们在理论上证明了在这种设置下可以缓解过度平滑的问题。在实验中，我们通过比较各种已建立的整数阶连续 GNN 的分数适应性，验证了 FROND 框架的有效性，并突出了该框架作为增强传统连续 GNN 的有效扩展的潜力。代码可在 https://github.com/zknus/ICLR2024-FROND 获得。