Typical distances in a geometric model for complex networks

We study typical distances in a geometric random graph on the hyperbolic plane. Introduced by Krioukov et al. as a model for complex networks, N vertices are drawn randomly within a bounded subset of the hyperbolic plane and any two of them are joined if they are within a threshold hyperbolic distance. With appropriately chosen parameters, the random graph is sparse and exhibits power law degree distribution as well as local clustering. In this paper we show a further property: the distance between two uniformly chosen vertices that belong to the same component is doubly logarithmic in N, i.e., the graph is an ultra-small world. More precisely, we show that the distance rescaled by loglog N converges in probability to a certain constant that depends on the exponent of the power law. The same constant emerges in an analogous setting with the well-known Chung-Lu model for which the degree distribution has a power law tail.