A Fractional Graph Laplacian Approach to Oversmoothing

TL;DR

This paper introduces fractional graph Laplacian neural ODEs to address oversmoothing in GNNs, enabling better long-range dependency capture in directed and undirected graphs through non-local dynamics.

Contribution

It extends oversmoothing concepts to directed graphs using a directed symmetrically normalized Laplacian and proposes fractional graph Laplacian neural ODEs for improved information propagation.

Findings

Effective in capturing long-range dependencies

Reduces oversmoothing in diverse graph types

Demonstrates versatility across homophily levels

Abstract

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 .