Title: Two Sides of the Same Coin: Heterophily and Oversmoothing in Graph Convolutional Neural Networks. Abstract: In node classification tasks, heterophily and oversmoothing are two problems that can hurt the performance of graph convolutional neural networks (GCNs). The heterophily problem refers to the model's inability to handle heterophilous graphs where neighboring nodes belong to different classes; the oversmoothing problem refers to the model's degenerated performance with increasing number of layers. These two seemingly unrelated problems have been studied mostly independently, but there is recent empirical evidence that solving one problem may benefit the other. 

In this work, beyond empirical observations, we aim to: (1) analyze the heterophily and oversmoothing problems from a unified theoretical perspective, (2) identify the common causes of the two problems based on our theories, and (3) propose simple yet effective strategies to address the common causes. In our theoretical analysis, we show that the common causes of the heterophily and oversmoothing problems---namely, the relative degree of a node (compared to its neighbors) and its heterophily level---trigger the node representations in consecutive layers to "move" closer to the original decision boundary, which increases the misclassification rate of node labels under certain constraints. We theoretically show that: (1) Nodes with high heterophily have a higher misclassification rate. (2) Even with low heterophily, degree disparity in a node's neighborhood can influence the movements of node representations and result in a "pseudo-heterophily" situation, which helps to explain oversmoothing. (3) Allowing not only positive, but also negative messages during message passing can help counteract the common causes of the two problems. Based on our theoretical insights, we propose simple modifications to the GCN architecture (i.e., learned degree corrections and signed messages), and we show that they alleviate the heteorophily and oversmoothing problems with extensive experiments on nine real networks. Compared to other approaches, which tend to work well in either heterophily or oversmoothing, our modified GCN model performs well in both problems.