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THE DEGREE PROFILE AND GINI INDEX OF RANDOM CATERPILLAR TREES

Published online by Cambridge University Press:  27 December 2018

Panpan Zhang  and
Dipak K. Dey
Panpan Zhang
Affiliation:
Department of Statistics, University of Connecticut, 215 Glenbrook Road U-4120, Storrs, CT 06269-4120, USA E-mail: panpan.zhang@uconn.edu; dipak.dey@uconn.edu
Dipak K. Dey
Affiliation:
Department of Statistics, University of Connecticut, 215 Glenbrook Road U-4120, Storrs, CT 06269-4120, USA E-mail: panpan.zhang@uconn.edu; dipak.dey@uconn.edu
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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 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.

Keywords

Degree profileGini indexMonte–Carlo experimentPólya urn modelrandom caterpillar treesstochastic recurrence
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 33 , Issue 4 , October 2019 , pp. 511 - 527
Copyright
Copyright © Cambridge University Press 2018 

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