  We introduce and explore a new method for inferring hidden geometric
coordinates of nodes in complex networks based on the number of common
neighbors between the nodes. We compare this approach to the HyperMap method,
which is based only on the connections (and disconnections) between the nodes,
i.e., on the links that the nodes have (or do not have). We find that for high
degree nodes the common-neighbors approach yields a more accurate inference
than the link-based method, unless heuristic periodic adjustments (or
"correction steps") are used in the latter. The common-neighbors approach is
computationally intensive, requiring $O(t^4)$ running time to map a network of
$t$ nodes, versus $O(t^3)$ in the link-based method. But we also develop a
hybrid method with $O(t^3)$ running time, which combines the common-neighbors
and link-based approaches, and explore a heuristic that reduces its running
time further to $O(t^2)$, without significant reduction in the mapping
accuracy. We apply this method to the Autonomous Systems (AS) Internet, and
reveal how soft communities of ASes evolve over time in the similarity space.
We further demonstrate the method's predictive power by forecasting future
links between ASes. Taken altogether, our results advance our understanding of
how to efficiently and accurately map real networks to their latent geometric
spaces, which is an important necessary step towards understanding the laws
that govern the dynamics of nodes in these spaces, and the fine-grained
dynamics of network connections.
