Planar growth generates scale free networks

TL;DR

This paper introduces a spatial network growth model that produces scale-free, highly clustered, small-world networks by enforcing planarity, and explores how relaxing planarity affects degree distributions.

Contribution

It presents a novel planar growth model with power law degree distribution, connecting it to Apollonian networks and analyzing the impact of planarity relaxation.

Findings

The model generates networks with power law degree distribution, high clustering, and small-world properties.

Relaxing planarity leads to a transition from power law to exponential degree distributions.

The model bridges random and deterministic Apollonian networks as special cases.

Abstract

In this paper we introduce a model of spatial network growth in which nodes are placed at randomly selected locations on a unit square in $\mathbb{R}^2$, 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 the two defining features of the network formation procedure; growth and planarity conservation. We demonstrate that the model can be understood as a variant of random Apollonian growth and further propose a one parameter family of models with the Random Apollonian Network and the Deterministic Apollonian Network as extreme cases and our model as a midpoint between them. We then relax the planarity constraint by allowing edge crossings with some probability and find a smooth crossover from power law to exponential degree distributions when this probability is increased.