2 Preliminary. We use A E to denote the existence of an edge between node u and v, otherwise A

Graph homophily refers to the phenomenon that connected nodes tend to share similar characteristics. Understanding this concept and its related metrics is crucial for designing effective Graph Neural Networks (GNNs). The most widely used homophily metrics, such as edge or node homophily, quantify such "similarity" as label consistency across the graph topology. These metrics are believed to be able to reflect the performance of GNNs, especially on node-level tasks. However, many recent studies have empirically demonstrated that the performance of GNNs does not always align with homophily metrics, and how homophily influences GNNs still remains unclear and controversial.