Graph neural networks (GNNs) are the 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）是图机器学习领域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研究工作，尤其关注其表征与外推能力。我们通过回答如下问题为GNN的表征与外推能力提供了新颖的理论视角：当图节点数量趋于无穷大时，GNN的行为如何？在温和假设条件下，我们证明：当从Erdős–Rényi模型中抽取规模递增的图时，此类图被某类GNN分类器映射到特定输出的概率趋于零或一。这类模型包括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，它蕴含了其能力上的理论局限性。我们通过实验验证了这一结论，观察到理论渐近极限在相对较小的图上已明显存在。