论文标题
图形卷积网络在大图随机图上的收敛性和稳定性
Convergence and Stability of Graph Convolutional Networks on Large Random Graphs
论文作者
论文摘要
我们通过分析其在随机图的标准模型上分析其行为来研究图形卷积网络（GCN）的属性，其中节点由随机潜在变量表示，并根据相似性内核绘制边缘。这使我们能够通过考虑更自然的几何方面来克服在非常大图上处理离散概念（例如同构）的困难。我们首先研究了随着节点的数量的增长，GCN与它们连续的对应物的收敛性。我们的结果是完全非肿瘤的，对于平均程度的相对稀疏图，与节点数量相对生长。然后，我们分析GCN对随机图模型的小变形的稳定性。与先前关于离散设置稳定性的研究相反，我们的连续设置使我们能够提供更直观的基于变形的指标来理解稳定性，这已证明对解释欧几里得领域上的卷积表示成功很有用。
We study properties of Graph Convolutional Networks (GCNs) by analyzing their behavior on standard models of random graphs, where nodes are represented by random latent variables and edges are drawn according to a similarity kernel. This allows us to overcome the difficulties of dealing with discrete notions such as isomorphisms on very large graphs, by considering instead more natural geometric aspects. We first study the convergence of GCNs to their continuous counterpart as the number of nodes grows. Our results are fully non-asymptotic and are valid for relatively sparse graphs with an average degree that grows logarithmically with the number of nodes. We then analyze the stability of GCNs to small deformations of the random graph model. In contrast to previous studies of stability in discrete settings, our continuous setup allows us to provide more intuitive deformation-based metrics for understanding stability, which have proven useful for explaining the success of convolutional representations on Euclidean domains.
