Graph neural networks (GNNs) have emerged as a powerful tool for nonlinear
processing of graph signals, exhibiting success in recommender systems, power
outage prediction, and motion planning, among others. GNNs consists of a
cascade of layers, each of which applies a graph convolution, followed by a
pointwise nonlinearity. In this work, we study the impact that changes in the
underlying topology have on the output of the GNN. First, we show that GNNs are
permutation equivariant, which implies that they effectively exploit internal
symmetries of the underlying topology. Then, we prove that graph convolutions
with integral Lipschitz filters, in combination with the frequency mixing
effect of the corresponding nonlinearities, yields an architecture that is both
stable to small changes in the underlying topology and discriminative of
information located at high frequencies. These are two properties that cannot
simultaneously hold when using only linear graph filters, which are either
discriminative or stable, thus explaining the superior performance of GNNs.

研究了拓扑结构变化对图神经网络输出的影响，证明了使用积分 Lipschitz 过滤器的图卷积与相应非线性混合效应的组合可以实现稳定性和高频信息的判别力，解释了 GNNs 卓越性能的原因。