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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
Affiliation:
Department of Statistics, University of Connecticut, 215 Glenbrook Road U-4120, Storrs, CT 06269-4120, USA E-mail: [email protected]; [email protected]
Dipak K. Dey
Affiliation:
Department of Statistics, University of Connecticut, 215 Glenbrook Road U-4120, Storrs, CT 06269-4120, USA E-mail: [email protected]; [email protected]

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.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2018 

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References

1Allen, S. & O'Donnell, R. (2015). Conditioning and covariance on caterpillars. In Proceedings of IEEE Information Theory Workshop, Jerusalem, Israel, pp. 15.Google Scholar
2Balaji, H. & Mahmoud, H. (2017). The Gini index of random trees with an application to caterpillars. Journal of Applied Probability 54: 701709.Google Scholar
3Barabási, A.-L. & Albert, R. (1999). Emergence of scaling in random networks. Science 286: 509512.Google Scholar
4Doob, J. (1990). Stochastic processes. New York: John Wiley & Sons, Inc..Google Scholar
5Eggenberger, F. & Pólya, G. (1923). Über die statistik verketteter vorgänge. Zeitschrift für Angewandte Mathematik und Mechanik 3: 279289.Google Scholar
6El-Basil, S. (1987). Applications of caterpillar trees in chemistry and physics. Journal of Mathematical Chemistry 1: 153174.Google Scholar
7Fisz, M. (1955). The limiting distribution of the multivariate distribution. Studia Mathematica 14: 272275.Google Scholar
8Gastwirth, J. (1972). The estimation of the Lorenz Curve and Gini index. The Review of Economics and Statistics 54: 306316.Google Scholar
9Goswami, S., Murthy, C., & Das, A. (2016). Sparsity measure of a network graph: Gini Index. ArXiv:1612.07074 [cs.DM].Google Scholar
10Gouet, R. (1997). Strong convergence of proportions in a multicolor Pólya urn. Journal of Applied Probability 34: 426435.Google Scholar
11Graczyk, P. (2007). Gini coefficient: a new way to express selectivity of kinase inhibitors against a family of kinases. Journal of Medicinal Chemistry 15: 57735779.Google Scholar
12Graham, R., Knuth, D., & Patashnik, O. (1989). Concrete mathematics. Reading, MA: Addison-Wesley Publishing Company.Google Scholar
13Hall, P. & Heyde, C. (1980). Martingale limit theory and its application. New York: Academic Press, Inc..Google Scholar
14Harary, F. & Schwenk, A. (1973). The number of caterpillars. Discrete Mathematics 6: 359365.Google Scholar
15Hobbs, A. (1973). Some Hamiltonian results in powers of graphs. Journal of Research of the National Bureau of Standards, Section B: Mathematics and Mathematical Physics 77B: 110.Google Scholar
16Hu, H.-B. & Wang, X.-F. (2008). Unified index to quantifying heterogeneity of complex networks. Physica A: Statistical Mechanics and its Applications 387: 37693780.Google Scholar
17Jamison, R., McMorris, F., & Mulder, H. (2003). Graphs with only caterpillars as spanning trees. Discrete Mathematics 272: 8195.Google Scholar
18Kennedy, B.P., Kawachi, I., Glass, R., & Prothrow-Stith, D. (2008). Income distribution, socioeconomic status, and self rated health in the United States: multilevel analysis. British Medicine Journal 317: 917921.Google Scholar
19Lee, W.-C. (1997). Characterizing exposure-disease association in human populations using the Lorenz curve and Gini index. Statistics in Medicine 16: 729739.Google Scholar
20Lerman, R. & Yitzhaki, S. (1984). A note on the calculation and interpretation of the Gini index. Economic Letters 15: 363368.Google Scholar
21Mahmoud, H. (2009). Pólya urn models. Boca Raton, FL: CRC Press.Google Scholar
22Marshall, A. & Olkin, I. (1990). Bivariate distributions generated from Pólya–Eggenberger urn models. Journal of Multivariate Analysis 3: 4865.Google Scholar
23Massé, A., de Carufel, J., Goupil, A., Lapointe, M., Nadeau, É., & Vandomme, É. (2017). Leaf realization problem, caterpillar graphs and prefix normal words. ArXiv:1712.01942 [math.CO].Google Scholar
24Miller, Z. (1981). The bandwidth of caterpillar graphs. In Proceedings of The 12th Southeastern Conference on Combinatorics, Graph Theory and Computing. Baton Rouge, LA, Congressus Numerantium 33: 235252.Google Scholar
25Musiela, M. & Rutkowski, M. (2005). Martingale methods in financial modelling. Berlin: Springer-Verlag.Google Scholar
26Ogwang, T. (2000). A convenient method of computing the Gini Index and its standard error. Oxford Bulletin of Economics and Statistics 62: 123129.Google Scholar
27Raychaudhuri, A. (1995). The total interval number of a tree and the Hamiltonian completion number of its line graph. Information Processing Letters 56: 299306.Google Scholar
28Roe, B., Yang, H.-J., Zhu, J., Liu, Y., Stancu, I., & McGregor, G. (2005). Boosted decision trees as an alternative to artificial neural networks for particle identification. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 543: 577584.Google Scholar
29Yule, G. (1925). A mathematical theory of evolution, based on the conclusions of Dr. J. C. Willis, F.R.S.. Philosophical Transactions of the Royal Society B 213: 2187.Google Scholar