Most graph convolutional neural networks (GCNs) perform poorly in graphs where neighbors typically have different features/classes (heterophily) and when stacking multiple layers (oversmoothing). These two seemingly unrelated problems have been studied independently, but there is recent empirical evidence that solving one problem may benefit the other. In this work, going beyond empirical observations, we aim to: (1) propose a new perspective to analyze the heterophily and oversmoothing problems under a unified theoretical framework, (2) identify the common causes of the two problems based on the proposed framework, and (3) propose simple yet effective strategies that address the common causes. Focusing on the node classification task, we use linear separability of node representations as an indicator to reflect the performance of GCNs and we propose to study the linear separability by analyzing the statistical change of the node representations in the graph convolution. We find that the relative degree of a node (compared to its neighbors) and the heterophily level of a nodes neighborhood are the root causes that influence the separability of node representations. Our analysis suggests that: (1) Nodes with high heterophily always produce less separable representations after graph convolution; (2) Even with low heterophily, degree disparity between nodes can influence the network dynamics and result in a pseudo-heterophily situation, which helps to explain oversmoothing. Based on our insights, we propose simple modifications to the GCN architecture -- i.e., degree corrections and signed messages -- which alleviate the root causes of these issues, and also show this empirically on 9 real networks. Compared to other approaches, which tend to work well in one regime but fail in others, our modified GCN model consistently performs well across all settings.
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