Spatial networks evolving to reduce length

TL;DR

This paper introduces a general scheme for evolving spatial networks to minimize total edge lengths, analyzing equilibria that interpolate between random graphs, geometric graphs, and spanning trees, with applications in social and transportation networks.

Contribution

It proposes a novel framework for evolving spatial networks to reduce length, connecting well-studied network models and exploring their properties and applications.

Findings

Analyzed equilibria between Erdős-Rényi, geometric, and spanning tree networks.

Developed models for social networks with fixed opinions adjusting ties.

Explored applications in transportation and distribution systems.

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 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.