Generative models play an important role in analyzing and predicting relevant characteristics
of real-world complex networks. The predictive power of these models lies in the possibility
that large amount of synthetic networks, similar to real ones at some extent, can
be produced, and by analyzing them statistically, significant inferences can be performed.
Synthetic networks, based on the hyperbolic geometry, turned out to be good generative
models of real-world networks, as they reproduce several macroscopic behaviours of
real networks and can help in assessing the scalability of new network functions.
The original hyperbolic generative model by Krioukov et al. [1] lacks an important
property: it can not guarantee with 100% probability that the resulting synthetic
network is connected. In other words for some parameter regions, the networks fall
into fragments resulting in disconnected regions of subgraphs. Inevitably, from the
viewpoint of modeling real networks, the connectedness is an expected property. In
this paper we show that, with a slight extension of the original generation rule of
the hyperbolic networks, the connectedness can always be ensured. We also present
analytical and numerical results which show that there will be no significant changes
in the macroscopic properties of networks, like average degree and degree distribution.
