  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 the so-called complex networks. The model is controlled by two
parameters $\alpha$ and $\nu$ where, roughly speaking, $\alpha$ controls the
exponent of the power law and $\nu$ 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
$\alpha,\nu$, while all other components are sublinear. We also study how $c$
depends on $\alpha, \nu$. To deduce our results, we introduce a local
approximation of the random graph by a continuum percolation model on
$\mathbb{R}^2$ that may be of independent interest.
