The degree profile and Gini index of random caterpillar trees

TL;DR

This paper analyzes the degree profile and Gini index of random caterpillar trees, providing exact distributions, asymptotic behavior, and a new index to differentiate growth mechanisms.

Contribution

It introduces methods to compute exact and asymptotic degree distributions and proposes a novel Gini index for comparing different RCT growth models.

Findings

Degree profiles follow multinomial and Dirichlet distributions.

Exact expectations and dispersion matrices are derived for degree variables.

A new Gini index effectively distinguishes uniform and preferential attachment RCTs.

Abstract

In this paper, we investigate the degree profile and Gini index of random caterpillar trees (RCTs). We consider RCTs which evolve in two different manners: uniform and nonuniform. The degrees of the vertices on the central path (i.e., the degree profile) of a uniform RCT follow a multinomial distribution. For nonuniform RCTs, we focus on those growing in the fashion of preferential attachment. We develop methods based on stochastic recurrences to compute the exact expectations and the dispersion matrix of the degree variables. A generalized P\'{o}lya urn model is exploited to determine the exact joint distribution of these degree variables. We apply the methods from combinatorics to prove that the asymptotic distribution is Dirichlet. In addition, we propose a new type of Gini index to quantitatively distinguish the evolutionary characteristics of the two classes of RCTs. We present the results via several numerical experiments.