  Most social, technological and biological networks are embedded in a finite
dimensional space, and the distance between two nodes influences the likelihood
that they link to each other. Indeed, in social systems, the chance that two
individuals know each other drops rapidly with the distance between them; in
the cell, proteins predominantly interact with proteins in the same cellular
compartment; in the brain, neurons mainly link to nearby neurons. Most modeling
frameworks that aim to capture the empirically observed degree distributions
tend to ignore these spatial constraints. In contrast, models that account for
the role of the physical distance often predict bounded degree distributions,
in disagreement with the empirical data. Here we address a long-standing gap in
the spatial network literature by deriving several key network characteristics
of spatial networks, from the analytical form of the degree distribution to
path lengths and local clustering. The mathematically exact results predict the
existence of two distinct phases, each governed by a different dynamical
equation, with distinct testable predictions. We use empirical data to offer
direct evidence for the practical relevance of each of these phases in real
networks, helping better characterize the properties of spatial networks.
