  Finding the optimal embedding of networks into low-dimensional hyperbolic
spaces is a challenge that received considerable interest in recent years, with
several different approaches proposed in the literature. In general, these
methods take advantage of the exponentially growing volume of the hyperbolic
space as a function of the radius from the origin, allowing a (roughly) uniform
spatial distribution of the nodes even for scale-free small-world networks,
where the connection probability between pairs decays with hyperbolic distance.
One of the motivations behind hyperbolic embedding is that optimal placement of
the nodes in a hyperbolic space is widely thought to enable efficient
navigation on top of the network. According to that, one of the measures that
can be used to quantify the quality of different embeddings is given by the
fraction of successful greedy paths following a simple navigation protocol
based on the hyperbolic coordinates. In the present work, we develop an
optimisation scheme for this score in the native disk representation of the
hyperbolic space. This optimisation algorithm can be either used as an
embedding method alone, or it can be applied to improve this score for
embeddings obtained from other methods. According to our tests on synthetic and
real networks, the proposed optimisation can considerably enhance the success
rate of greedy paths in several cases, improving the given embedding from the
point of view of navigability.
