  Random hyperbolic graphs were recently introduced by Krioukov et. al.
[KPKVB10] as a model for large networks. Gugelmann, Panagiotou, and Peter
[GPP12] then initiated the rigorous study of random hyperbolic graphs using the
following model: for $\alpha> \tfrac{1}{2}$, $C\in\mathbb{R}$,
$n\in\mathbb{N}$, set $R=2\ln n+C$ and build the graph $G=(V,E)$ with $|V|=n$
as follows: For each $v\in V$, generate i.i.d. polar coordinates
$(r_{v},\theta_{v})$ using the joint density function $f(r,\theta)$, with
$\theta_{v}$ chosen uniformly from $[0,2\pi)$ and $r_{v}$ with density
$f(r)=\frac{\alpha\sinh(\alpha r)}{\cosh(\alpha R)-1}$ for $0\leq r< R$. Then,
join two vertices by an edge, if their hyperbolic distance is at most $R$. We
prove that in the range $\tfrac{1}{2} < \alpha < 1$ a.a.s. for any two vertices
of the same component, their graph distance is $O(\log^{C_0+1+o(1)}n)$, where
$C_0=2/(\tfrac{1}{2}-\frac{3}{4}\alpha+\tfrac{\alpha^2}{4})$, thus answering a
question raised in [GPP12] concerning the diameter of such random graphs. As a
corollary from our proof we obtain that the second largest component has size
$O(\log^{2C_0+1+o(1)}n)$, thus answering a question of Bode, Fountoulakis and
M\"{u}ller [BFM13]. We also show that a.a.s. there exist isolated components
forming a path of length $\Omega(\log n)$, thus yielding a lower bound on the
size of the second largest component.
