Graph neural networks (GNNs) have shown state-of-the-art performances in various applications. However, GNNs often struggle to capture long-range dependencies in graphs due to oversmoothing. In this paper, we generalize the concept of oversmoothing from undirected to directed graphs. To this aim, we extend the notion of Dirichlet energy by considering a directed symmetrically normalized Laplacian. As vanilla graph convolutional networks are prone to oversmooth, we adopt a neural graph ODE framework. Specifically, we propose fractional graph Laplacian neural ODEs, which describe non-local dynamics. We prove that our approach allows propagating information between distant nodes while maintaining a low probability of long-distance jumps. Moreover, we show that our method is more flexible with respect to the convergence of the graph's Dirichlet energy, thereby mitigating oversmoothing. We conduct extensive experiments on synthetic and real-world graphs, both directed and undirected, demonstrating our method's versatility across diverse graph homophily levels. Our code is available at https://github.com/RPaolino/fLode .

翻译：图神经网络（GNNs）在各种应用中展现出最先进的性能。然而，由于过度平滑问题，GNNs 通常难以捕捉图中的长程依赖关系。本文中，我们将过度平滑的概念从无向图推广到有向图。为此，我们通过考虑有向对称归一化拉普拉斯算子，扩展了狄利克雷能量的概念。由于传统图卷积网络容易出现过平滑现象，我们采用神经图 ODE 框架。具体而言，我们提出了分数阶图拉普拉斯神经 ODE，其描述了非局部动力学。我们证明，该方法能够在远距离节点之间传播信息，同时保持较低的长距离跳跃概率。此外，我们表明，所提方法在图狄利克雷能量的收敛性方面更加灵活，从而缓解了过度平滑问题。我们在合成图和真实世界图（包括有向图和无向图）上进行了广泛实验，证明了我们的方法在不同图同质性水平下的通用性。我们的代码可在 https://github.com/RPaolino/fLode 获取。