Abstract
Hyperbolic models are remarkably good at reproducing the scale-free, highly clustered and smallworld properties of networks representing real complex systems in a very simple framework. Here we show that for the popularity-similarity optimization model from this family, the generated networks become also extremely modular in the thermodynamic limit, in spite of lacking any explicit community formation mechanism in the model definition. According to our analytical results supported by numerical simulations, when the system size is increased, the modularity approaches one surprisingly fast.