We characterize the performance of graph neural networks for graph alignment problems in the presence of vertex feature information. More specifically, given two graphs that are independent perturbations of a single random geometric graph with noisy sparse features, the task is to recover an unknown one-to-one mapping between the vertices of the two graphs. We show under certain conditions on the sparsity and noise level of the feature vectors, a carefully designed one-layer graph neural network can with high probability recover the correct alignment between the vertices with the help of the graph structure. We also prove that our conditions on the noise level are tight up to logarithmic factors. Finally we compare the performance of the graph neural network to directly solving an assignment problem on the noisy vertex features. We demonstrate that when the noise level is at least constant this direct matching fails to have perfect recovery while the graph neural network can tolerate noise level growing as fast as a power of the size of the graph.

翻译：我们刻画了在顶点特征信息存在时，图神经网络在图对齐问题中的性能表现。具体而言，给定两个独立扰动自同一随机几何图且带有噪声稀疏特征的图，任务是恢复两个图顶点之间未知的一一映射关系。我们证明：在特征向量的稀疏性和噪声水平满足特定条件时，精心设计的单层图神经网络能够以高概率借助图结构恢复顶点间的正确对齐。同时，我们证明了关于噪声水平的条件在对数因子意义下是紧的。最后，我们将图神经网络的性能与直接求解含噪顶点特征分配问题的方法进行了比较。我们证明：当噪声水平至少为常数时，直接匹配方法无法实现完美恢复，而图神经网络能够容忍以图大小的幂次增长的噪声水平。