  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 Bollob\'as 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.
