  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.
