  Whilst spectral Graph Neural Networks (GNNs) are theoretically well-founded
in the spectral domain, their practical reliance on polynomial approximation
implies a profound linkage to the spatial domain. As previous studies rarely
examine spectral GNNs from the spatial perspective, their spatial-domain
interpretability remains elusive, e.g., what information is essentially encoded
by spectral GNNs in the spatial domain? In this paper, to answer this question,
we investigate the theoretical connection between spectral filtering and
spatial aggregation, unveiling an intrinsic interaction that spectral filtering
implicitly leads the original graph to an adapted new graph, explicitly
computed for spatial aggregation. Both theoretical and empirical investigations
reveal that the adapted new graph not only exhibits non-locality but also
accommodates signed edge weights to reflect label consistency among nodes.
These findings thus highlight the interpretable role of spectral GNNs in the
spatial domain and inspire us to rethink graph spectral filters beyond the
fixed-order polynomials, which neglect global information. Built upon the
theoretical findings, we revisit the state-of-the-art spectral GNNs and propose
a novel Spatially Adaptive Filtering (SAF) framework, which leverages the
adapted new graph by spectral filtering for an auxiliary non-local aggregation.
Notably, our SAF comprehensively models both node similarity and dissimilarity
from a global perspective, therefore alleviating persistent deficiencies of
GNNs related to long-range dependencies and graph heterophily. Extensive
experiments over 13 node classification benchmarks demonstrate the superiority
of our proposed framework to the state-of-the-art methods.
