Graph Convolutional Neural Networks (graph CNNs) are a promising deep learning approach for analyzing graph-structured data. However, it is known that they do not improve (or sometimes worsen) their predictive performance as we pile up more layers and make them deeper. To tackle this problem, we investigate the expressive power of graph CNNs by analyzing their asymptotic behaviors as the layer size tends to infinity. Our strategy is to generalize the forward propagation of a Graph Convolutional Network (GCN), which is one of the most popular graph CNN variants, as a specific dynamical system. In the case of GCNs, we show that when the weights satisfy the conditions determined by the spectra of the (augmented) normalized Laplacian, the output of GCNs exponentially approaches the set of signals that carry only information of the connected components and node degrees for distinguishing nodes. Our theory enables us to directly relate the expressive power of GCNs with the topological information of the underlying graphs, which is inherent in the graph spectra. To demonstrate this, we characterize the asymptotic behavior of GCNs on the Erd\H{o}s -- R\'{e}nyi graph. We show that when the Erd\H{o}s -- R\'{e}nyi graph is sufficiently dense and large, a wide range of GCNs on them suffers from this ``information loss" in the limit of infinite layers with high probability. Furthermore, our theory provides principled guidelines for the weight normalization of graph CNNs. We experimentally confirmed that weight scaling based on our theory enhanced the predictive performance of GCNs in real data.

对图神经网络进行研究，通过看其在无穷大层神经元的情况下的渐进行为，探讨了GCNs的表达能力，发现GCNs的表达能力与其底层图的拓扑信息相关，提供了一种基于权值归一化的方法，增强了真实数据下GCNs的性能。

图神经网络在节点分类方面逐渐失去表达能力