Convergence and Stability of Graph Convolutional Networks on Large Random Graphs
This work addresses theoretical understanding of GCN behavior for machine learning researchers, offering incremental insights by extending stability analysis from discrete to continuous settings.
The paper tackles the analysis of Graph Convolutional Networks (GCNs) on large random graphs by studying their convergence to continuous counterparts and stability to deformations, providing non-asymptotic results for sparse graphs with logarithmic average degree growth.
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.