  Graph Neural Networks (GNNs) have demonstrated strong performance in graph
mining tasks due to their message-passing mechanism, which is aligned with the
homophily assumption that adjacent nodes exhibit similar behaviors. However, in
many real-world graphs, connected nodes may display contrasting behaviors,
termed as heterophilous patterns, which has attracted increased interest in
heterophilous GNNs (HTGNNs). Although the message-passing mechanism seems
unsuitable for heterophilous graphs due to the propagation of class-irrelevant
information, it is still widely used in many existing HTGNNs and consistently
achieves notable success. This raises the question: why does message passing
remain effective on heterophilous graphs? To answer this question, in this
paper, we revisit the message-passing mechanisms in heterophilous graph neural
networks and reformulate them into a unified heterophilious message-passing
(HTMP) mechanism. Based on HTMP and empirical analysis, we reveal that the
success of message passing in existing HTGNNs is attributed to implicitly
enhancing the compatibility matrix among classes. Moreover, we argue that the
full potential of the compatibility matrix is not completely achieved due to
the existence of incomplete and noisy semantic neighborhoods in real-world
heterophilous graphs. To bridge this gap, we introduce a new approach named
CMGNN, which operates within the HTMP mechanism to explicitly leverage and
improve the compatibility matrix. A thorough evaluation involving 10 benchmark
datasets and comparative analysis against 13 well-established baselines
highlights the superior performance of the HTMP mechanism and CMGNN method.
