Distances in random Apollonian network structures

TL;DR

This paper analyzes the distribution of distances in random Apollonian networks, revealing that the average distance between vertices grows asymptotically with the square root of the network size.

Contribution

It introduces a generating function approach and singularity analysis to describe distance distributions in RANS, a novel method for this network family.

Findings

Average distance asymptotically proportional to square root of n

Distribution of distances to outermost vertex characterized

Method applicable to other planar network analyses

Abstract

In this paper, we study the distribution of distances in random Apollonian network structures (RANS), a family of graphs which has a one-to-one correspondence with planar ternary trees. Using multivariate generating functions that express all information on distances, and singularity analysis for evaluating the coefficients of these functions, we describe the distribution of distances to an outermost vertex, and show that the average value of the distance between any pair of vertices in a RANS of order n is asymptotically square root of n.