  In node classification tasks, graph convolutional neural networks (GCNs) have
demonstrated competitive performance over traditional methods on diverse graph
data. However, it is known that the performance of GCNs degrades with
increasing number of layers (oversmoothing problem) and recent studies have
also shown that GCNs may perform worse in heterophilous graphs, where
neighboring nodes tend to belong to different classes (heterophily problem).
These two problems are usually viewed as unrelated, and thus are studied
independently, often at the graph filter level from a spectral perspective.
  We are the first to take a unified perspective to jointly explain the
oversmoothing and heterophily problems at the node level. Specifically, we
profile the nodes via two quantitative metrics: the relative degree of a node
(compared to its neighbors) and the node-level heterophily. Our theory shows
that the interplay of these two profiling metrics defines three cases of node
behaviors, which explain the oversmoothing and heterophily problems jointly and
can predict the performance of GCNs. Based on insights from our theory, we show
theoretically and empirically the effectiveness of two strategies:
structure-based edge correction, which learns corrected edge weights from
structural properties (i.e., degrees), and feature-based edge correction, which
learns 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 both
problems.
