  Link prediction is a paradigmatic problem in network science with a variety
of applications. In latent space network models this problem boils down to
ranking pairs of nodes in the order of increasing latent distances between
them. The network model with hyperbolic latent spaces has a number of
attractive properties suggesting it must be a powerful tool to predict links,
but the past work in this direction reported mixed results. Here we perform
systematic investigation of the utility of latent hyperbolic geometry for link
prediction in networks. We first show that some measures of link prediction
accuracy are extremely sensitive with respect to inaccuracies in the inference
of latent hyperbolic coordinates of nodes, so that we develop a new coordinate
inference method that maximizes the accuracy of such inference. Applying this
method to synthetic and real networks, we then find that while there exists a
multitude of competitive methods to predict obvious easy-to-predict links,
among which hyperbolic link prediction is rarely the best but often
competitive, it is the best, often by far, when the task is to predict less
obvious missing links that are really hard to predict. These links include
missing links in incomplete networks with large fractions of missing links,
missing links between nodes that do not have any common neighbors, and missing
links between dissimilar nodes at large latent distances. Overall these results
suggest that the harder a specific link prediction task is, the more seriously
one should consider using hyperbolic geometry.
