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Homophily principle, i.e., nodes with the same labels are more likely to beconnected, has been believed to be the main reason for the performancesuperiority of Graph Neural Networks (GNNs) over Neural Networks on nodeclassification tasks. Recent research suggests that, even in the absence ofhomophily, the advantage of GNNs still exists as long as nodes from the sameclass share similar neighborhood patterns. However, this argument onlyconsiders intra-class Node Distinguishability (ND) but neglects inter-class ND,which provides incomplete understanding of homophily on GNNs. In this paper, wefirst demonstrate such deficiency with examples and argue that an idealsituation for ND is to have smaller intra-class ND than inter-class ND. Toformulate this idea and study ND deeply, we propose Contextual Stochastic BlockModel for Homophily (CSBM-H) and define two metrics, Probabilistic Bayes Error(PBE) and negative generalized Jeffreys divergence, to quantify ND. With themetrics, we visualize and analyze how graph filters, node degree distributionsand class variances influence ND, and investigate the combined effect of intra-and inter-class ND. Besides, we discovered the mid-homophily pitfall, whichoccurs widely in graph datasets. Furthermore, we verified that, in real-worktasks, the superiority of GNNs is indeed closely related to both intra- andinter-class ND regardless of homophily levels. Grounded in this observation, wepropose a new hypothesis-testing based performance metric beyond homophily,which is non-linear, feature-based and can provide statistical threshold valuefor GNNs' the superiority. Experiments indicate that it is significantly moreeffective than the existing homophily metrics on revealing the advantage anddisadvantage of graph-aware modes on both synthetic and benchmark real-worlddatasets.

When Do Graph Neural Networks Help with Node Classification? Investigating the Impact of Homophily Principle on Node Distinguishability