  Graph Neural Networks (GNNs) have received extensive research attention for
their promising performance in graph machine learning. Despite their
extraordinary predictive accuracy, existing approaches, such as GCN and GPRGNN,
are not robust in the face of homophily changes on test graphs, rendering these
models vulnerable to graph structural attacks and with limited capacity in
generalizing to graphs of varied homophily levels. Although many methods have
been proposed to improve the robustness of GNN models, most of these techniques
are restricted to the spatial domain and employ complicated defense mechanisms,
such as learning new graph structures or calculating edge attentions. In this
paper, we study the problem of designing simple and robust GNN models in the
spectral domain. We propose EvenNet, a spectral GNN corresponding to an
even-polynomial graph filter. Based on our theoretical analysis in both spatial
and spectral domains, we demonstrate that EvenNet outperforms full-order models
in generalizing across homophilic and heterophilic graphs, implying that
ignoring odd-hop neighbors improves the robustness of GNNs. We conduct
experiments on both synthetic and real-world datasets to demonstrate the
effectiveness of EvenNet. Notably, EvenNet outperforms existing defense models
against structural attacks without introducing additional computational costs
and maintains competitiveness in traditional node classification tasks on
homophilic and heterophilic graphs.
