  Hyperbolic network models, centered around the idea of placing nodes at
random in a hyperbolic space and drawing links according to a probability that
decreases as a function of the distance, provide a simple, yet also very
capable framework for grasping the small-world, scale-free, highly clustered
and modular nature of complex systems that are often referred to as real-world
networks. In the present work we study the community structure of networks
generated by the Popularity Similarity Optimization model (corresponding to one
of the fundamental, widely known hyperbolic models) when the temperature
parameter (responsible for tuning the clustering coefficient) is set to the
limiting value of zero. By focusing on the intra-community link formation we
derive analytical expressions for the expected modularity of a partitioning
consisting of equally sized angular sectors in the native disk representation
of the 2d hyperbolic space. Our formulas improve earlier results to a great
extent, being able to estimate the average modularity (measured by numerical
simulations) with high precision in a considerably larger range both in terms
of the model parameters and also the relative size of the communities with
respect to the entire network. These findings enhance our comprehension of how
modules form in hyperbolic networks. The existence of these modules is somewhat
unexpected, given the absence of explicit community formation steps in the
model definition.
