  Spectral Graph Neural Networks (GNNs) are gaining attention for their ability
to surpass the limitations of message-passing GNNs. They rely on supervision
from downstream tasks to learn spectral filters that capture the graph signal's
useful frequency information. However, some works empirically show that the
preferred graph frequency is related to the graph homophily level. This
relationship between graph frequency and graphs with homophily/heterophily has
not been systematically analyzed and considered in existing spectral GNNs. To
mitigate this gap, we conduct theoretical and empirical analyses revealing a
positive correlation between low-frequency importance and the homophily ratio,
and a negative correlation between high-frequency importance and the homophily
ratio. Motivated by this, we propose shape-aware regularization on a Newton
Interpolation-based spectral filter that can (i) learn an arbitrary polynomial
spectral filter and (ii) incorporate prior knowledge about the desired shape of
the corresponding homophily level. Comprehensive experiments demonstrate that
NewtonNet can achieve graph spectral filters with desired shapes and superior
performance on both homophilous and heterophilous datasets.
