Apollonian networks

TL;DR

This paper introduces Apollonian networks, a new family of networks that are scale-free, small-world, and Euclidean, with diverse applications and unique properties such as rich clustering and peculiar percolation and conduction behaviors.

Contribution

The paper presents the first detailed study of Apollonian networks, highlighting their construction, properties, and potential applications across various fields.

Findings

Networks are scale-free and small-world.

Rich clustering coefficient and connectivity exponent.

Unique behaviors in percolation and electrical conduction.

Abstract

We introduce a new family of networks, the Apollonian networks, that are simultaneously scale-free, small world, Euclidean, space-filling and matching graphs. These networks have a wide range of applications ranging from the description of force chains in polydisperse granular packings and geometry of fully fragmented porous media, to hierarchical road systems and area-covering electrical supply networks. Some of the properties of these networks, namely, the connectivity exponent, the clustering coefficient, and the shortest path are calculated and found to be particularly rich. The percolation, the electrical conduction and the Ising models on such networks are also studied and found to be quite peculiar. Consequences for applications are also discussed.