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

We introduce the FRactional-Order graph Neural Dynamical network (FROND), a learning framework that extends traditional graph neural ordinary differential equation (ODE) models by incorporating the time-fractional Caputo derivative. Due to its non-local nature, fractional calculus allows our framework to capture long-term memories in the feature updating process, in contrast to the Markovian nature of updates in traditional graph neural ODE models. This can lead to improved graph representation learning. We offer an interpretation of the feature updating process on graphs from a non-Markovian random walk perspective when the feature updating is governed by a diffusion process. We demonstrate analytically that over-smoothing can be mitigated in this setting. To experimentally demonstrate the versatility of the FROND framework, we evaluate the fractional counterparts of various established graph ODE models. Their consistently superior performance, compared to their original counterparts, highlights the potential of the FROND framework as an effective extension to boost the efficacy of various graph neural ODE models.