Graph neural networks (GNNs) are de facto standard deep learning architectures for machine learning on graphs. This has led to a large body of work analyzing the capabilities and limitations of these models, particularly pertaining to their representation and extrapolation capacity. We offer a novel theoretical perspective on the representation and extrapolation capacity of GNNs, by answering the question: how do GNNs behave as the number of graph nodes become very large? Under mild assumptions, we show that when we draw graphs of increasing size from the Erd\H{o}s-R\'enyi model, the probability that such graphs are mapped to a particular output by a class of GNN classifiers tends to either zero or to one. This class includes the popular graph convolutional network architecture. The result establishes 'zero-one laws' for these GNNs, and analogously to other convergence laws, entails theoretical limitations on their capacity. We empirically verify our results, observing that the theoretical asymptotic limits are evident already on relatively small graphs.

翻译：图神经网络（GNNs）是面向图数据的机器学习事实标准深度学习架构。这促使大量研究工作分析这类模型的表达能力与局限性，特别是其表征能力和外推能力。我们通过回答以下问题为GNN的表征与外推能力提供全新理论视角：当图节点数量变得极大时，GNN会呈现何种行为？在温和假设下，我们证明：当从 Erdős–Rényi 模型中抽取规模递增的图时，一类GNN分类器将这些图映射到特定输出的概率趋于零或一。该分类器类包含流行的图卷积网络架构。该结果确立了这些GNN的"零一律"，与其他收敛定律类似，揭示了其容量的理论限制。我们通过实验验证了理论结果，观察到理论渐近极限在规模相对较小的图上已经显现。