What Is Still Missing in Graph Homophily? Degree Regularity Matters

Graph homophily, the tendency of connected nodes to share similar characteristics, is widely used to understand and predict the performance of Graph Neural Networks (GNNs). Most homophily metrics, however, capture only local similarity patterns, such as label or feature consistency across edges. In this paper, we study whether a global graph property, the dispersion of the degree distribution, carries performance-relevant information that local homophily misses. We introduce degree regularity, which quantifies how uniform the node degrees are, relate degree dispersion to aggregation noise through elementary results, and add it to a local homophily factor to obtain degree-aware glocal homophily. To study degree dispersion in a controlled setting, we develop a calibrated degree-aware synthetic graph generator, which reveals that varying degree dispersion can silently vary density; at nearly fixed density, the effect of dispersion on GNN accuracy changes sign with label homophily, as a stylized aggregation model explains. On 31 real-world datasets, with baseline metrics recomputed from their definitions, degree regularity is positively associated with GNN accuracy beyond local homophily. Degree-aware glocal homophily attains the highest correlation with the accuracy of GNNs among 24 metrics. These results suggest that global degree properties complement local homophily in explaining GNN performance.