We suggest to model human-annotated Word Usage Graphs capturing fine-grained semantic proximity distinctions between word uses with a Bayesian formulation of the Weighted Stochastic Block Model, a generative model for random graphs popular in biology, physics and social sciences. By providing a probabilistic model of graded word meaning we aim to approach the slippery and yet widely used notion of word sense in a novel way. The proposed framework enables us to rigorously compare models of word senses with respect to their fit to the data. We perform extensive experiments and select the empirically most adequate model.