  In this paper we consider the clustering coefficient and clustering function
in a random graph model proposed by Krioukov et al.~in 2010. In this model,
nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes
are connected if they are at most a certain hyperbolic distance from each
other. It has been shown that this model has various properties associated with
complex networks, e.g. power-law degree distribution, short distances and
non-vanishing clustering coefficient. Here we show that the clustering
coefficient tends in probability to a constant $\gamma$ that we give explicitly
as a closed form expression in terms of $\alpha, \nu$ and certain special
functions. This improves earlier work by Gugelmann et al., who proved that the
clustering coefficient remains bounded away from zero with high probability,
but left open the issue of convergence to a limiting constant. Similarly, we
are able to show that $c(k)$, the average clustering coefficient over all
vertices of degree exactly $k$, tends in probability to a limit $\gamma(k)$
which we give explicitly as a closed form expression in terms of $\alpha, \nu$
and certain special functions. We are able to extend this last result also to
sequences $(k_n)_n$ where $k_n$ grows as a function of $n$. Our results show
that $\gamma(k)$ scales differently, as $k$ grows, for different ranges of
$\alpha$. More precisely, there exists constants $c_{\alpha,\nu}$ depending on
$\alpha$ and $\nu$, such that as $k \to \infty$, $\gamma(k) \sim c_{\alpha,\nu}
\cdot k^{2 - 4\alpha}$ if $\frac{1}{2} < \alpha < \frac{3}{4}$, $\gamma(k) \sim
c_{\alpha,\nu} \cdot \log(k) \cdot k^{-1} $ if $\alpha=\frac{3}{4}$ and
$\gamma(k) \sim c_{\alpha,\nu} \cdot k^{-1}$ when $\alpha > \frac{3}{4}$. These
results contradict a claim of Krioukov et al., which stated that the limiting
values $\gamma(k)$ should always scale with $k^{-1}$ as we let $k$ grow.
