  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 .
