Graph convolutional networks (GCNs) are a widely used method for graph representation learning. We investigate the power of GCNs, as a function of their number of layers, to distinguish between different random graph models on the basis of the embeddings of their sample graphs. In particular, the graph models that we consider arise from graphons, which are the most general possible parameterizations of infinite exchangeable graph models and which are the central objects of study in the theory of dense graph limits. We exhibit an infinite class of graphons that are well-separated in terms of cut distance and are indistinguishable by a GCN with nonlinear activation functions coming from a certain broad class if its depth is at least logarithmic in the size of the sample graph. These results theoretically match empirical observations of several prior works. Finally, we show a converse result that for pairs of graphons satisfying a degree profile separation property, a very simple GCN architecture suffices for distinguishability. To prove our results, we exploit a connection to random walks on graphs.

翻译：图形图变图网络(GCNs) 是一种广泛使用的图形演示学习方法。 我们调查了GCNs的力量, 以其层数的函数来区分不同随机图形模型, 其依据是样品图的嵌入。 特别是, 我们所考虑的图形模型来自图形, 它们是无限可交换的图形模型的最一般可能的参数, 也是密度图形限制理论中研究的中心对象。 我们展示了无穷的几类图形子, 它们以切开距离的方式分离, 并且无法分辨, 其非线性激活功能来自某大类的GCN, 如果其深度至少与样本图的大小有对数。 这些结果在理论上匹配了先前若干作品的经验性观测。 最后, 我们展示了一个对应结果, 对符合度剖析图属性属性的相配方来说, 一个非常简单的GCN结构足以辨别。 为了证明我们的结果, 我们探索了图表上随机行的连接。