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

翻译：同质性原则，即相同标签的节点更倾向于连接，一直被认为是图神经网络（GNN）在节点分类任务上优于神经网络的性能优势的主要原因。近期研究表明，即使缺乏同质性，只要来自同一类别的节点共享相似的邻域模式，GNN的这一优势依然存在。然而，该论点仅考虑了类内节点可区分性（ND），却忽视了类间ND，导致对GNN中同质性的理解不完整。本文首先通过示例论证了这一缺陷，并提出ND的理想状态是类内ND小于类间ND。为形式化这一观点并深入研究ND，我们提出了适用于同质性的上下文随机块模型（CSBM-H），并定义了概率贝叶斯误差（PBE）和负广义杰弗里斯散度两个度量指标来量化ND。利用这些指标，我们可视化并分析了图滤波器、节点度分布和类方差如何影响ND，并探究了类内与类间ND的联合效应。此外，我们发现了广泛存在于图数据集中的中同质性陷阱。进一步验证表明，在实际任务中，无论同质性水平如何，GNN的优势确实与类内和类间ND密切相关。基于这一观察，我们提出了一种超越同质性的基于假设检验的性能度量，该度量是非线性的、基于特征的，并能提供GNN优势的统计阈值。实验表明，在揭示图感知模式在合成和基准真实世界数据集上的优缺点方面，该度量显著优于现有同质性度量。