  Motivated by results of Henry, Pralat and Zhang (PNAS 108.21 (2011):
8605-8610), we propose a general scheme for evolving spatial networks in order
to reduce their total edge lengths. We study the properties of the equilbria of
two networks from this class, which interpolate between three well studied
objects: the Erd\H{o}s-R\'{e}nyi random graph, the random geometric graph, and
the minimum spanning tree. The first of our two evolutions can be used as a
model for a social network where individuals have fixed opinions about a number
of issues and adjust their ties to be connected to people with similar views.
The second evolution which preserves the connectivity of the network has
potential applications in the design of transportation networks and other
distribution systems.
