  Reducing dimension redundancy to find simplifying patterns in
high-dimensional datasets and complex networks has become a major endeavor in
many scientific fields. However, detecting the dimensionality of their latent
space is challenging but necessary to generate efficient embeddings to be used
in a multitude of downstream tasks. Here, we propose a method to infer the
dimensionality of networks without the need for any a priori spatial embedding.
Due to the ability of hyperbolic geometry to capture the complex connectivity
of real networks, we detect ultra low dimensionality far below values reported
using other approaches. We applied our method to real networks from different
domains and found unexpected regularities, including: tissue-specific
biomolecular networks being extremely low dimensional; brain connectomes being
close to the three dimensions of their anatomical embedding; and social
networks and the Internet requiring slightly higher dimensionality. Beyond
paving the way towards an ultra efficient dimensional reduction, our findings
help address fundamental issues that hinge on dimensionality, such as
universality in critical behavior.
