Generative algorithms for random graphs have yielded insights into the structure and evolution of real-world networks. Most networks exhibit a well-known set of properties, such as heavy-tailed degree distributions, clustering and community formation. Usually, random graph models consider only structural information, but many real-world networks also have labelled vertices and weighted edges. In this paper, we present a generative model for random graphs with discrete vertex labels and numeric edge weights. The weights are represented as a set of Beta Mixture Models (BMMs) with an arbitrary number of mixtures, which are learned from real-world networks. We propose a Bayesian Variational Inference (VI) approach, which yields an accurate estimation while keeping computation times tractable. We compare our approach to state-of-the-art random labelled graph generators and an earlier approach based on Gaussian Mixture Models (GMMs). Our results allow us to draw conclusions about the contribution of vertex labels and edge weights to graph structure.