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 article, 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 nonlocality but also accommodates signed edge weights to reflect label consistency among nodes. These findings 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 limit the effective propagation range and hinder their ability to capture long-range dependencies. 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 nonlocal 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.
Guo et al. (Thu,) studied this question.