From Local to Global: Spectral-Inspired Graph Neural Networks

TL;DR

This paper introduces PowerEmbed, a spectral-inspired normalization layer for GNNs that enhances their ability to capture both local and global graph signals, addressing issues like over-smoothing and over-squashing.

Contribution

The paper proposes PowerEmbed, a simple spectral-inspired normalization technique that provably captures top eigenvectors and improves GNN performance on various graph types.

Findings

PowerEmbed can express top-k eigenvectors of the graph operator.

It prevents over-smoothing and over-squashing in GNNs.

Demonstrates competitive performance on heterophilous and real-world graphs.

Abstract

Graph Neural Networks (GNNs) are powerful deep learning methods for Non-Euclidean data. Popular GNNs are message-passing algorithms (MPNNs) that aggregate and combine signals in a local graph neighborhood. However, shallow MPNNs tend to miss long-range signals and perform poorly on some heterophilous graphs, while deep MPNNs can suffer from issues like over-smoothing or over-squashing. To mitigate such issues, existing works typically borrow normalization techniques from training neural networks on Euclidean data or modify the graph structures. Yet these approaches are not well-understood theoretically and could increase the overall computational complexity. In this work, we draw inspirations from spectral graph embedding and propose $\texttt{PowerEmbed}$ -- a simple layer-wise normalization technique to boost MPNNs. We show $\texttt{PowerEmbed}$ can provably express the top-$k$ leading eigenvectors of the graph operator, which prevents over-smoothing and is agnostic to the graph topology; meanwhile, it produces a list of representations ranging from local features to global signals, which avoids over-squashing. We apply $\texttt{PowerEmbed}$ in a wide range of simulated and real graphs and demonstrate its competitive performance, particularly for heterophilous graphs.