Spatial Message Passing Graph Neural Networks (MPGNNs) are widely used for learning on graph-structured data. However, key limitations of l-step MPGNNs are that their "receptive field" is typically limited to the l-hop neighborhood of a node and that information exchange between distant nodes is limited by over-squashing. Motivated by these limitations, we propose Spatio-Spectral Graph Neural Networks (S$^2$GNNs) -- a new modeling paradigm for Graph Neural Networks (GNNs) that synergistically combines spatially and spectrally parametrized graph filters. Parameterizing filters partially in the frequency domain enables global yet efficient information propagation. We show that S$^2$GNNs vanquish over-squashing and yield strictly tighter approximation-theoretic error bounds than MPGNNs. Further, rethinking graph convolutions at a fundamental level unlocks new design spaces. For example, S$^2$GNNs allow for free positional encodings that make them strictly more expressive than the 1-Weisfeiler-Lehman (WL) test. Moreover, to obtain general-purpose S$^2$GNNs, we propose spectrally parametrized filters for directed graphs. S$^2$GNNs outperform spatial MPGNNs, graph transformers, and graph rewirings, e.g., on the peptide long-range benchmark tasks, and are competitive with state-of-the-art sequence modeling. On a 40 GB GPU, S$^2$GNNs scale to millions of nodes.

翻译：空间消息传递图神经网络（MPGNNs）被广泛用于图结构数据的学习。然而，l步MPGNNs的关键局限在于其“感受野”通常仅限于节点的l跳邻域，且远距离节点间的信息交换受限于过度挤压。受这些局限性的启发，我们提出了空谱图神经网络（S$^2$GNNs）——一种新的图神经网络（GNNs）建模范式，它协同结合了空间参数化和谱参数化的图滤波器。在频域中部分参数化滤波器能够实现全局且高效的信息传播。我们证明S$^2$GNNs克服了过度挤压问题，并产生了比MPGNNs严格更紧的逼近论误差界。此外，从根本上重新思考图卷积解锁了新的设计空间。例如，S$^2$GNNs允许使用自由位置编码，使其表达能力严格强于1-Weisfeiler-Lehman（WL）测试。而且，为了获得通用目的的S$^2$GNNs，我们提出了针对有向图的谱参数化滤波器。S$^2$GNNs在性能上超越了空间MPGNNs、图Transformer和图重连方法，例如在肽长程基准任务上，并且与最先进的序列建模方法具有竞争力。在40 GB GPU上，S$^2$GNNs可扩展至数百万节点。