Law of large numbers for the largest component in a hyperbolic model of complex networks

We consider the component structure of a recent model of random graphs on the hyperbolic plane that was introduced by Krioukov et al. The model exhibits a power law degree sequence, small distances and clustering, features that are associated with so-called complex networks. The model is controlled by two parameters α and ν where, roughly speaking, α controls the exponent of the power law and ν controls the average degree. Refining earlier results, we are able to show a law of large numbers for the largest component. That is, we show that the fraction of points in the largest component tends in probability to a constant c that depends only on α,ν, while all other components are sublinear. We also study how c depends on α,ν. To deduce our results, we introduce a local approximation of the random graph by a continuum percolation model on ℝ2 that may be of independent interest.