2018/05/15 by Panpan Zhang, Dipak K. Dey · 9 citations
Mathematics · Physics and Astronomy · #Applied mathematics #Combinatorics #Complex Network Analysis Techniques #Computer science #Degree (music) #Degree distribution #Dirichlet distribution #Distribution (mathematics) #Index (typography) #Mathematical analysis #Mathematics #Multinomial distribution #Statistics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.ST #stat.CO #stat.TH
paper · pdf · doi:10.1017/s0269964818000475
published in Probability in the Engineering and Informational Sciences 33(4), 511-527 (Cambridge University Press)
arxiv created 2018/05/15 · openalex publication_date 2018/12/27 · arxiv updated 2019/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.