We investigate the classification performance of graph neural networks with graph-polynomial features, poly-GNNs, on the problem of semi-supervised node classification. We analyze poly-GNNs under a general contextual stochastic block model (CSBM) by providing a sharp characterization of the rate of separation between classes in their output node representations. A question of interest is whether this rate depends on the depth of the network $k$, i.e., whether deeper networks can achieve a faster separation? We provide a negative answer to this question: for a sufficiently large graph, a depth $k > 1$ poly-GNN exhibits the same rate of separation as a depth $k=1$ counterpart. Our analysis highlights and quantifies the impact of ``graph noise'' in deep GNNs and shows how noise in the graph structure can dominate other sources of signal in the graph, negating any benefit further aggregation provides. Our analysis also reveals subtle differences between even and odd-layered GNNs in how the feature noise propagates.

翻译：本文研究了具有图多项式特征的图神经网络（poly-GNNs）在半监督节点分类问题上的分类性能。我们在一般上下文随机块模型（CSBM）下分析poly-GNNs，通过精确刻画其输出节点表示中类别间分离速率的特征。一个值得关注的问题是：该速率是否依赖于网络深度$k$，即更深的网络能否实现更快的分离？我们对此给出了否定答案：对于足够大的图，深度$k > 1$的poly-GNN表现出与深度$k=1$对应模型相同的分离速率。我们的分析突出并量化了深度GNN中“图噪声”的影响，揭示了图结构中的噪声如何主导图中其他信号源，从而抵消进一步聚合带来的任何优势。分析还揭示了偶数层与奇数层GNN在特征噪声传播方式上的微妙差异。