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Graph Neural Networks (GNNs) achieve state-of-the-art performance ongraph-structured data across numerous domains. Their underlying ability torepresent nodes as summaries of their vicinities has proven effective forhomophilous graphs in particular, in which same-type nodes tend to connect. Onheterophilous graphs, in which different-type nodes are likely connected, GNNsperform less consistently, as neighborhood information might be lessrepresentative or even misleading. On the other hand, GNN performance is notinferior on all heterophilous graphs, and there is a lack of understanding ofwhat other graph properties affect GNN performance.  In this work, we highlight the limitations of the widely used homophily ratioand the recent Cross-Class Neighborhood Similarity (CCNS) metric in estimatingGNN performance. To overcome these limitations, we introduce 2-hop NeighborClass Similarity (2NCS), a new quantitative graph structural property thatcorrelates with GNN performance more strongly and consistently than alternativemetrics. 2NCS considers two-hop neighborhoods as a theoretically derivedconsequence of the two-step label propagation process governing GCN'straining-inference process. Experiments on one synthetic and eight real-worldgraph datasets confirm consistent improvements over existing metrics inestimating the accuracy of GCN- and GAT-based architectures on the nodeclassification task.

2-hop Neighbor Class Similarity (2NCS): A graph structural metric indicative of graph neural network performance