  We investigate graph neural networks on graphs with heterophily. Some
existing methods amplify a node's neighborhood with multi-hop neighbors to
include more nodes with homophily. However, it is a significant challenge to
set personalized neighborhood sizes for different nodes. Further, for other
homophilous nodes excluded in the neighborhood, they are ignored for
information aggregation. To address these problems, we propose two models
GloGNN and GloGNN++, which generate a node's embedding by aggregating
information from global nodes in the graph. In each layer, both models learn a
coefficient matrix to capture the correlations between nodes, based on which
neighborhood aggregation is performed. The coefficient matrix allows signed
values and is derived from an optimization problem that has a closed-form
solution. We further accelerate neighborhood aggregation and derive a linear
time complexity. We theoretically explain the models' effectiveness by proving
that both the coefficient matrix and the generated node embedding matrix have
the desired grouping effect. We conduct extensive experiments to compare our
models against 11 other competitors on 15 benchmark datasets in a wide range of
domains, scales and graph heterophilies. Experimental results show that our
methods achieve superior performance and are also very efficient.
