  A remarkable approach for grasping the relevant statistical features of real
networks with the help of random graphs is offered by hyperbolic models,
centred around the idea of placing nodes in a low-dimensional hyperbolic space,
and connecting node pairs with a probability depending on the hyperbolic
distance. It is widely appreciated that these models can generate random graphs
that are small-world, highly clustered and scale-free at the same time; thus,
reproducing the most fundamental common features of real networks. In the
present work, we focus on a less well-known property of the
popularity-similarity optimisation (PSO) model and the
$\mathbb{S}^1/\mathbb{H}^2$ model from this model family, namely that the
networks generated by these approaches also contain communities for a wide
range of the parameters, which was certainly not an intention at the design of
the models. We extracted the communities from the studied networks using
well-established community finding methods such as Louvain, Infomap and label
propagation. The observed high modularity values indicate that the community
structure can become very pronounced under certain conditions. In addition, the
modules found by the different algorithms show good consistency, implying that
these are indeed relevant and apparent structural units. Since the appearance
of communities is rather common in networks representing real systems as well,
this feature of hyperbolic models makes them even more suitable for describing
real networks than thought before.
