The modularity of random graphs on the hyperbolic plane

Modularity is a quantity which has been introduced in the context of complex networks in order to quantify how close a network is to an ideal modular network in which the nodes form small interconnected communities that are joined together with relatively few edges. In this article, we consider this quantity on a probabilistic model of complex networks introduced by Krioukov et al. (2010, Phys. Rev. E, 82, 036106). This model views a complex network as an expression of hidden popularity hierarchies (i.e. nodes higher up in the hierarchies have more global reach), encapsulated by an underlying hyperbolic space. For certain parameters, this model was proved to have typical features that are observed in complex networks such as power law degree distribution, bounded average degree, clustering coefficient that is asymptotically bounded away from zero and ultra-small typical distances. In the present work, we investigate its modularity and we show that, in this regime, it converges to one in probability