What Mean Aggregation Retains in GNNs: Degree, Precision, and Learned Readouts

Mean aggregation in GNNs is non-injective, but this does not imply that it discards all information about neighborhood size. We characterize when degree remains recoverable from the aggregate and ask whether learned GNNs exploit that information. For categorical messages, the exact mean consists of empirical proportions (counts divided by degree), whose denominators constrain, and can sometimes identify, neighborhood size. Rounding or noise can remove this degree signal while preserving information about neighborhood composition. We use Fisher-information analysis to show how degree information depends on the message distribution: for Gaussian messages, the retained fraction vanishes at high degree even as information about the message mean remains. A broad class of regular readouts recovers asymptotically none of the degree information retained by exact categorical means; fitted neural readouts exhibit the same gap, and reducing numerical precision barely changes trained-GNN predictions. Supplying degree explicitly improves log loss and robustness under a low-degree shift but provides little benefit at high degree. Together, these results show that mean aggregation can retain degree information without learned GNNs making effective use of it.