THE DEGREE PROFILE AND GINI INDEX OF RANDOM CATERPILLAR TREES

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 follows a multinomial distri...

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Bibliographic Details
Published inProbability in the engineering and informational sciences Vol. 33; no. 4; pp. 511 - 527
Main Authors Zhang, Panpan, Dey, Dipak K.
Format Journal Article
LanguageEnglish
Published New York, USA Cambridge University Press 01.10.2019
Subjects
Apexes
Asymptotic methods
Combinatorial analysis
Dirichlet problem
Trees
random caterpillar trees
Monte–Carlo experiment
Gini index
stochastic recurrence
Degree profile
Pólya urn model
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Summary: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 follows 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 Pólya 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.
ISSN:0269-9648
1469-8951
DOI:10.1017/S0269964818000475