Spatial networks evolving to reduce length
Chris Varghese and Rick Durrett
Abstract.
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 Erdos-Renyi 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.
Preprint
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