General model for Apollonian networks

TL;DR

This paper presents a versatile deterministic model for Apollonian networks that exhibits small-world and scale-free properties, with exact calculations for key network metrics showing adjustable degree exponents and logarithmic diameter growth.

Contribution

It introduces a general iterative model for Apollonian networks with exact analytical results for key topological properties, enhancing understanding of their structure.

Findings

Degree exponent can be tuned over a wide range.

Clustering coefficient inversely proportional to degree.

Network diameter grows logarithmically with size.

Abstract

We introduce a general deterministic model for Apollonian Networks in an iterative fashion. The networks have small-world effect and scale-free topology. We calculate the exact results for the degree exponent, the clustering coefficient and the diameter. The major points of our results indicate that (a) the degree exponent can be adjusted in a wide range, (b) the clustering coefficient of each individual vertex is inversely proportional to its degree and the average clustering coefficient of all vertices approaches to a nonzero value in the infinite network order, and (c) the diameter grows logarithmically with the number of network vertices.