  We introduce Mercator, a reliable embedding method to map real complex
networks into their hyperbolic latent geometry. The method assumes that the
structure of networks is well described by the Popularity$\times$Similarity
$\mathbb{S}^1/\mathbb{H}^2$ static geometric network model, which can
accommodate arbitrary degree distributions and reproduces many pivotal
properties of real networks, including self-similarity patterns. The algorithm
mixes machine learning and maximum likelihood approaches to infer the
coordinates of the nodes in the underlying hyperbolic disk with the best
matching between the observed network topology and the geometric model. In its
fast mode, Mercator uses a model-adjusted machine learning technique performing
dimensional reduction to produce a fast and accurate map, whose quality already
outperform other embedding algorithms in the literature. In the refined
Mercator mode, the fast-mode embedding result is taken as an initial condition
in a Maximum Likelihood estimation, which significantly improves the quality of
the final embedding. Apart from its accuracy as an embedding tool, Mercator has
the clear advantage of systematically inferring not only node orderings, or
angular positions, but also the hidden degrees and global model parameters, and
has the ability to embed networks with arbitrary degree distributions. Overall,
our results suggest that mixing machine learning and maximum likelihood
techniques in a model-dependent framework can boost the meaningful mapping of
complex networks.
