Characterizing several properties of high-dimensional random Apollonian networks

TL;DR

This paper rigorously analyzes various structural properties of high-dimensional random Apollonian networks, including degree distributions, small-world characteristics, sparsity, and distance metrics, using advanced mathematical and probabilistic methods.

Contribution

It provides a comprehensive mathematical characterization of multiple properties of HDRANs, employing novel analytical techniques and measures.

Findings

Degree profiles characterized by probabilistic methods

Small-world property confirmed via local clustering coefficient

Sparsity assessed with a new Gini index

Abstract

In this article, we investigate several properties of high-dimensional random Apollonian networks (HDRANs), including two types of degree profiles, the small-world effect (clustering property), sparsity, and three distance-based metrics. The characterizations of degree profiles are based on several rigorous mathematical and probabilistic methods, such as a two-dimensional mathematical induction, analytic combinatorics, and P\'{o}lya urns, etc. The small-world property is uncovered by a well-developed measure---local clustering coefficient, and the sparsity is assessed by a proposed Gini index. Finally, we look into three distance-based properties; they are total depth, diameter and Wiener index.