  The fundamental idea of embedding a network in a metric space is rooted in
the principle of proximity preservation. Nodes are mapped into points of the
space with pairwise distance that reflects their proximity in the network.
Popular methods employed in network embedding either rely on implicit
approximations of the principle of proximity preservation or implement it by
enforcing the geometry of the embedding space, thus hindering geometric
properties that networks may spontaneously exhibit. Here, we take advantage of
a model-free embedding method explicitly devised for preserving pairwise
proximity, and characterize the geometry emerging from the mapping of several
networks, both real and synthetic. We show that the learned embedding has
simple and intuitive interpretations: the distance of a node from the geometric
center is representative for its closeness centrality, and the relative
positions of nodes reflect the community structure of the network. Proximity
can be preserved in relatively low-dimensional embedding spaces, and the hidden
geometry displays optimal performance in guiding greedy navigation regardless
of the specific network topology. We finally show that the mapping provides a
natural description of contagion processes on networks, with complex
spatiotemporal patterns represented by waves propagating from the geometric
center to the periphery. The findings deepen our understanding of the
model-free hidden geometry of complex networks.
