  Large real-world networks are typically scale-free. Recent research has shown
that such graphs are described best in a geometric space. More precisely, the
internet can be mapped to a hyperbolic space such that geometric greedy routing
performs close to optimal (Bogun\'a, Papadopoulos, and Krioukov. Nature
Communications, 1:62, 2010). This observation pushed the interest in hyperbolic
networks as a natural model for scale-free networks. Hyperbolic random graphs
follow a power-law degree distribution with controllable exponent $\beta$ and
show high clustering (Gugelmann, Panagiotou, and Peter. ICALP, pp. 573-585,
2012).
  For understanding the structure of the resulting graphs and for analyzing the
behavior of network algorithms, the next question is bounding the size of the
diameter. The only known explicit bound is $\mathcal O((\log
n)^{32/((3-\beta)(5-\beta)) + 1})$ (Kiwi and Mitsche. ANALCO, pp. 26-39, 2015).
We present two much simpler proofs for an improved upper bound of $\mathcal
O((\log n)^{2/(3-\beta)})$ and a lower bound of $\Omega(\log n)$.

Discrete Mathematics