Typical distances in a geometric model for complex networks

The theme of this paper is the study of typical distances in a ran-dom graph model that was introduced by Krioukov et al. [25] and envisages basic properties of complex networks as the expression of an underlying hyperbolic geometry. This model gives rise to sparse ran-dom graphs on the hyperbolic plane which exhibit power law degree distribution as well as local clustering (i.e., they are locally dense). In this paper, we show that in fact these random graphs are ultra-small worlds. When the parameters of the model yield a power law degree distribution with exponent between 2 and 3, we show that the dis-tance between two given vertices conditional on belonging to the same component is proportional to log logN a.a.s., where N is the number of vertices of the random graph. To be more precise, we show that the distance rescaled by log logN converges in probability to a certain con-stant that depends on the exponent of the power law. This constant actually emerges in the same setting in the Chung-Lu model. We also consider the regime where the exponent of the power law is larger than 3. There we show that most pairs of vertices that belong to the same component are within distance poly(log logN) with high probability asymptotically as N →∞