  Recently, many carefully crafted graph representation learning methods have
achieved impressive performance on either strong heterophilic or homophilic
graphs, but not both. Therefore, they are incapable of generalizing well across
real-world graphs with different levels of homophily. This is attributed to
their neglect of homophily in heterophilic graphs, and vice versa. In this
paper, we propose a two-fold filtering mechanism to extract homophily in
heterophilic graphs and vice versa. In particular, we extend the graph heat
equation to perform heterophilic aggregation of global information from a long
distance. The resultant filter can be exactly approximated by the
Possion-Charlier (PC) polynomials. To further exploit information at multiple
orders, we introduce a powerful graph convolution PC-Conv and its instantiation
PCNet for the node classification task. Compared with state-of-the-art GNNs,
PCNet shows competitive performance on well-known homophilic and heterophilic
graphs. Our implementation is available at https://github.com/uestclbh/PC-Conv.
