Contact graphs of disk packings as a model of spatial planar networks

TL;DR

This paper introduces a minimal model for spatial maximal planar networks based on disk packings, providing analytical insights into their properties and comparing them with Apollonian networks to understand the impact of spatial constraints.

Contribution

The paper presents a new disk packing-based model for spatial planar networks with analytical solutions for key properties, highlighting the role of spatial constraints in network structure.

Findings

Model reproduces key features of real spatial networks

Analytical solutions for degree distribution, clustering, and path length

Structural differences linked to disk packing methods

Abstract

Spatially constrained planar networks are frequently encountered in real-life systems. In this paper, based on a space-filling disk packing we propose a minimal model for spatial maximal planar networks, which is similar to but different from the model for Apollonian networks [J. S. Andrade, Jr. et al., Phys. Rev. Lett. {\bf 94}, 018702 (2005)]. We present an exhaustive analysis of various properties of our model, and obtain the analytic solutions for most of the features, including degree distribution, clustering coefficient, average path length, and degree correlations. The model recovers some striking generic characteristics observed in most real networks. To address the robustness of the relevant network properties, we compare the structural features between the investigated network and the Apollonian networks. We show that topological properties of the two networks are encoded in the way of disk packing. We argue that spatial constrains of nodes are relevant to the structure of the networks.