From Local to Global: Spectral-Inspired Graph Neural Networks

Ningyuan Teresa Huang, Soledad Villar, Carey Priebe, Da Zheng, Chengyue Huang, Lin Yang, Vladimir Braverman

Keywords: graph neural networks, spectral embedding, community detection

TL;DR: Motivated from spectral graph embedding. we propose PowerEmbed, a simple layer-wise normalization technique to inject global information in message-passing graph neural networks.,

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.

Community Implementations: [ 1 code implementation](https://www.catalyzex.com/paper/arxiv:2209.12054/code)