  Networks in nature are often formed within a spatial domain in a dynamical
manner, gaining links and nodes as they develop over time. We propose a class
of spatially-based growing network models and investigate the relationship
between the resulting statistical network properties and the dimension and
topology of the space in which the networks are embedded. In particular, we
consider models in which nodes are placed one by one in random locations in
space, with each such placement followed by configuration relaxation toward
uniform node density, and connection of the new node with spatially nearby
nodes. We find that such growth processes naturally result in networks with
small-world features, including a short characteristic path length and nonzero
clustering. These properties do not appear to depend strongly on the topology
of the embedding space, but do depend strongly on its dimension;
higher-dimensional spaces result in shorter path lengths but less clustering.
