Deeper with Riemannian Geometry: Overcoming Oversmoothing and Oversquashing for Graph Foundation Models

TL;DR

This paper introduces a Riemannian geometry-inspired local approach to improve message passing neural networks, effectively addressing oversmoothing and oversquashing issues for deeper graph models.

Contribution

It proposes a novel local adjustment method using Riemannian geometry, with theoretical guarantees and superior performance on various graph types.

Findings

GBN maintains performance with over 256 layers

Theoretical link between spectral gap and gradient vanishing

Effective handling of oversmoothing and oversquashing

Abstract

Message Passing Neural Networks (MPNNs) is the building block of graph foundation models, but fundamentally suffer from oversmoothing and oversquashing. There has recently been a surge of interest in fixing both issues. Existing efforts primarily adopt global approaches, which may be beneficial in some regions but detrimental in others, ultimately leading to the suboptimal expressiveness. In this paper, we begin by revisiting oversquashing through a global measure -- spectral gap $\lambda$ -- and prove that the increase of $\lambda$ leads to gradient vanishing with respect to the input features, thereby undermining the effectiveness of message passing. Motivated by such theoretical insights, we propose a \textbf{local} approach that adaptively adjusts message passing based on local structures. To achieve this, we connect local Riemannian geometry with MPNNs, and establish a novel nonhomogeneous boundary condition to address both oversquashing and oversmoothing. Building on the Robin condition, we design a GBN network with local bottleneck adjustment, coupled with theoretical guarantees. Extensive experiments on homophilic and heterophilic graphs show the expressiveness of GBN. Furthermore, GBN does not exhibit performance degradation even when the network depth exceeds $256$ layers.