Unleashing the Potential of Fractional Calculus in Graph Neural Networks with FROND

TL;DR

This paper introduces FROND, a novel continuous graph neural network framework using fractional calculus, which captures long-term dependencies and mitigates oversmoothing, showing improved performance over traditional models.

Contribution

FROND is the first to incorporate fractional derivatives into continuous GNNs, enabling non-local feature updates and better long-term dependency modeling.

Findings

FROND improves performance of existing continuous GNNs.

Fractional calculus mitigates oversmoothing in GNNs.

Enhanced graph representation learning capabilities.

Abstract

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 \url{https://github.com/zknus/ICLR2024-FROND}.