  Graph Neural Networks (GNNs) exhibit excellent performance when graphs have
strong homophily property, i.e. connected nodes have the same labels. However,
they perform poorly on heterophilic graphs. Several approaches address the
issue of heterophily by proposing models that adapt the graph by optimizing
task-specific loss function using labelled data. These adaptations are made
either via attention or by attenuating or enhancing various
low-frequency/high-frequency signals, as needed for the task at hand. More
recent approaches adapt the eigenvalues of the graph. One important
interpretation of this adaptation is that these models select/weigh the
eigenvectors of the graph. Based on this interpretation, we present an
eigendecomposition based approach and propose EigenNetwork models that improve
the performance of GNNs on heterophilic graphs. Performance improvement is
achieved by learning flexible graph adaptation functions that modulate the
eigenvalues of the graph. Regularization of these functions via parameter
sharing helps to improve the performance even more. Our approach achieves up to
11% improvement in performance over the state-of-the-art methods on
heterophilic graphs.
