  While graph heterophily has been extensively studied in recent years, a
fundamental research question largely remains nascent: How and to what extent
will graph heterophily affect the prediction performance of graph neural
networks (GNNs)? In this paper, we aim to demystify the impact of graph
heterophily on GNN spectral filters. Our theoretical results show that it is
essential to design adaptive polynomial filters that adapts different degrees
of graph heterophily to guarantee the generalization performance of GNNs.
Inspired by our theoretical findings, we propose a simple yet powerful GNN
named GPatcher by leveraging the MLP-Mixer architectures. Our approach
comprises two main components: (1) an adaptive patch extractor function that
automatically transforms each node's non-Euclidean graph representations to
Euclidean patch representations given different degrees of heterophily, and (2)
an efficient patch mixer function that learns salient node representation from
both the local context information and the global positional information.
Through extensive experiments, the GPatcher model demonstrates outstanding
performance on node classification compared with popular homophily GNNs and
state-of-the-art heterophily GNNs.
