Since the Message Passing (Graph) Neural Networks (MPNNs) have a linear
complexity with respect to the number of nodes when applied to sparse graphs,
they have been widely implemented and still raise a lot of interest even though
their theoretical expressive power is limited to the first order
Weisfeiler-Lehman test (1-WL). In this paper, we show that if the graph
convolution supports are designed in spectral-domain by a non-linear custom
function of eigenvalues and masked with an arbitrary large receptive field, the
MPNN is theoretically more powerful than the 1-WL test and experimentally as
powerful as a 3-WL existing models, while remaining spatially localized.
Moreover, by designing custom filter functions, outputs can have various
frequency components that allow the convolution process to learn different
relationships between a given input graph signal and its associated properties.
So far, the best 3-WL equivalent graph neural networks have a computational
complexity in $\mathcal{O}(n^3)$ with memory usage in $\mathcal{O}(n^2)$,
consider non-local update mechanism and do not provide the spectral richness of
output profile. The proposed method overcomes all these aforementioned problems
and reaches state-of-the-art results in many downstream tasks.

本文研究了基于图传递的神经网络（MPNNs），通过在频域设计非线性的自定义滤波器函数并使用任意大的感受野进行掩膜，其理论上比 1-WL 测试显得更加强大，且实际上具有与现有 3-WL 模型同样的性能，同时仍保持局部化。通过本方法可以实现 LOD 更新机制和输出谱复杂性，从而在许多下游任务中达到先进水平。