Planarity as a driver of Spatial Network structure

Abstract

In this paper we introduce a new model of spatial network growth in which nodes are placed at randomly selected locations in space over time, forming new connections to old nodes subject to the constraint that edges do not cross. The resulting network has a power law degree distribution, high clustering and the small world property. We argue that these characteristics are a consequence of two features of our mechanism, growth and planarity conservation. We further propose that our model can be understood as a variant of random Apollonian growth. We then investigate the robustness of our findings by relaxing the planarity. Specifically, we allow edges to cross with a defined probability. Varying this probability demonstrates a smooth transition from a power law to an exponential degree distribution.