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In node classification tasks, graph convolutional neural networks (GCNs) havedemonstrated competitive performance over traditional methods on diverse graphdata. However, it is known that the performance of GCNs degrades withincreasing number of layers (oversmoothing problem) and recent studies havealso shown that GCNs may perform worse in heterophilous graphs, whereneighboring nodes tend to belong to different classes (heterophily problem).These two problems are usually viewed as unrelated, and thus are studiedindependently, often at the graph filter level from a spectral perspective.  We are the first to take a unified perspective to jointly explain theoversmoothing and heterophily problems at the node level. Specifically, weprofile the nodes via two quantitative metrics: the relative degree of a node(compared to its neighbors) and the node-level heterophily. Our theory showsthat the interplay of these two profiling metrics defines three cases of nodebehaviors, which explain the oversmoothing and heterophily problems jointly andcan predict the performance of GCNs. Based on insights from our theory, we showtheoretically and empirically the effectiveness of two strategies:structure-based edge correction, which learns corrected edge weights fromstructural properties (i.e., degrees), and feature-based edge correction, whichlearns signed edge weights from node features. Compared to other approaches,which tend to handle well either heterophily or oversmoothing, we show that{our model, GGCN}, which incorporates the two strategies performs well in bothproblems.

Two Sides of the Same Coin: Heterophily and Oversmoothing in Graph Convolutional Neural Networks