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（GNN）已在各类应用中展现出最先进的性能。然而，由于过平滑问题，GNN往往难以捕捉图中的长程依赖关系。本文从无向图推广到有向图，对过平滑概念进行了泛化。为此，我们通过引入有向对称归一化拉普拉斯算子，扩展了狄利克雷能量的定义。针对经典图卷积网络易出现过平滑的问题，我们采用了神经图ODE框架，具体提出了分数阶图拉普拉斯神经ODE，以描述非局部动力学特性。我们证明该方法能在保持低概率长距离跳跃的同时，实现远距离节点间的信息传播。此外，我们展示了所提方法在图的狄利克雷能量收敛性方面更具灵活性，从而有效缓解过平滑。在合成图与真实图（包括有向图与无向图）上开展的广泛实验表明，本方法在不同图同质性水平下均具有普适性。相关代码已开源：https://github.com/RPaolino/fLode。