L. Zdeborová

O. Duranthon, L. Zdeborová

The contextual stochastic block model (cSBM) was proposed for unsupervised community detection on attributed graphs where both the graph and the high-dimensional node information correlate with node labels. In the context of machine learning on graphs, the cSBM has been widely used as a synthetic dataset for evaluating the performance of graph-neural networks (GNNs) for semi-supervised node classification. We consider a probabilistic Bayes-optimal formulation of the inference problem and we derive a belief-propagation-based algorithm for the semi-supervised cSBM; we conjecture it is optimal in the considered setting and we provide its implementation. We show that there can be a considerable gap between the accuracy reached by this algorithm and the performance of the GNN architectures proposed in the literature. This suggests that the cSBM, along with the comparison to the performance of the optimal algorithm, readily accessible via our implementation, can be instrumental in the development of more performant GNN architectures.

O. Duranthon, L. Zdeborová

While graph convolutional networks show great practical promises, the theoretical understanding of their generalization properties as a function of the number of samples is still in its infancy compared to the more broadly studied case of supervised fully connected neural networks. In this article, we predict the performances of a single-layer graph convolutional network (GCN) trained on data produced by attributed stochastic block models (SBMs) in the high-dimensional limit. Previously, only ridge regression on contextual-SBM (CSBM) has been considered in Shi et al. 2022; we generalize the analysis to arbitrary convex loss and regularization for the CSBM and add the analysis for another data model, the neural-prior SBM. We also study the high signal-to-noise ratio limit, detail the convergence rates of the GCN and show that, while consistent, it does not reach the Bayes-optimal rate for any of the considered cases.

O. Duranthon, L. Zdeborová

Graph neural networks (GNNs) are designed to process data associated with graphs. They are finding an increasing range of applications; however, as with other modern machine learning techniques, their theoretical understanding is limited. GNNs can encounter difficulties in gathering information from nodes that are far apart by iterated aggregation steps. This situation is partly caused by so-called oversmoothing; and overcoming it is one of the practically motivated challenges. We consider the situation where information is aggregated by multiple steps of convolution, leading to graph convolutional networks (GCNs). We analyze the generalization performance of a basic GCN, trained for node classification on data generated by the contextual stochastic block model. We predict its asymptotic performance by deriving the free energy of the problem, using the replica method, in the high-dimensional limit. Calling depth the number of convolutional steps, we show the importance of going to large depth to approach the Bayes-optimality. We detail how the architecture of the GCN has to scale with the depth to avoid oversmoothing. The resulting large depth limit can be close to the Bayes-optimality and leads to a continuous GCN. Technically, we tackle this continuous limit via an approach that resembles dynamical mean-field theory (DMFT) with constraints at the initial and final times. An expansion around large regularization allows us to solve the corresponding equations for the performance of the deep GCN. This promising tool may contribute to the analysis of further deep neural networks.