  Graph Neural Network (GNN) research has highlighted a relationship between
high homophily (i.e., the tendency of nodes of the same class to connect) and
strong predictive performance in node classification. However, recent work has
found the relationship to be more nuanced, demonstrating that simple GNNs can
learn in certain heterophilous settings. To resolve these conflicting findings
and align closer to real-world datasets, we go beyond the assumption of a
global graph homophily level and study the performance of GNNs when the local
homophily level of a node deviates from the global homophily level. Through
theoretical and empirical analysis, we systematically demonstrate how shifts in
local homophily can introduce performance degradation, leading to performance
discrepancies across local homophily levels. We ground the practical
implications of this work through granular analysis on five real-world datasets
with varying global homophily levels, demonstrating that (a) GNNs can fail to
generalize to test nodes that deviate from the global homophily of a graph, and
(b) high local homophily does not necessarily confer high performance for a
node. We further show that GNNs designed for globally heterophilous graphs can
alleviate performance discrepancy by improving performance across local
homophily levels, offering a new perspective on how these GNNs achieve stronger
global performance.
