We consider a non-projective class of inhomogeneous random graph models with interpretable parameters and a number of interesting asymptotic properties. Using the results of Bollobás et al. [2007], we show that i) the class of models is sparse and ii) depending on the choice of the parameters, the model is either scale-free, with power-law exponent greater than 2, or with an asymptotic degree distribution which is power-law with exponential cut-off. We propose an extension of the model that can accommodate an overlapping community structure. Scalable posterior inference can be performed due to the specific choice of the link probability. We present experiments on five different real-world networks with up to 100,000 nodes and edges, showing that the model can provide a good fit to the degree distribution and recovers well the latent community structure.