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）是图数据机器学习中事实上的标准深度学习架构。这催生了大量分析这些模型能力与局限性的工作，尤其关注其表示能力与外推能力。我们通过回答以下问题对GNN的表示能力与外推能力提出了一种新颖的理论视角：当图节点数量变得非常大时，GNN如何表现？在温和假设下，我们证明：当从Erdős–Rényi模型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"，类似于其他收敛定律，这对其能力施加了理论限制。我们通过实验验证了结果，观察到理论渐近极限已在相对较小的图上显现。