  Hyperbolic network models have gained considerable attention in recent years,
mainly due to their capability of explaining many peculiar features of
real-world networks. One of the most widely known models of this type is the
popularity-similarity optimisation (PSO) model, working in the native disk
representation of the two-dimensional hyperbolic space and generating networks
with small-world property, scale-free degree distribution, high clustering and
strong community structure at the same time. With the motivation of better
understanding hyperbolic random graphs, we hereby introduce the $d$PSO model, a
generalisation of the PSO model to any arbitrary integer dimension $d>2$. The
analysis of the obtained networks shows that their major structural properties
can be affected by the dimension of the underlying hyperbolic space in a
non-trivial way. Our extended framework is not only interesting from a
theoretical point of view but can also serve as a starting point for the
generalisation of already existing two-dimensional hyperbolic embedding
techniques.
