Optimal Performance of Graph Convolutional Networks on the Contextual Stochastic Block Model

For Graph Neural Networks, oversmoothing denotes the homogenization of vertex embeddings as the number of layers increases. To better understand this phenomenon, we study community detection with a linearized Graph Convolutional Network on the Contextual Stochastic Block Model. We express the distribution of the embeddings in each community as a Gaussian mixture over a low-dimensional latent space, with explicit formulas in the case of a single layer. This yields tractable estimators for classification accuracy at finite depth. Numerical experiments suggest that modeling with a single Gaussian is insufficient and that the impact of depth may be more complex than previously anticipated.