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On Nodes of Small Degrees and Degree Profile in Preferential Dynamic Attachment Circuits

Author

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  • Panpan Zhang

    (Epidemiology and Informatics at the University of Pennsylvania)

  • Hosam M. Mahmoud

    (The George Washington University)

Abstract

We investigate the joint distribution of nodes of small degrees and the degree profile in preferential dynamic attachment circuits. In particular, we study the joint asymptotic distribution of the number of the nodes of outdegree 0 (terminal nodes) and outdegree 1 in a very large circuit. The expectation and variance of the number of those two types of nodes are both asymptotically linear with respect to the age of the circuit. We show that the numbers of nodes of outdegree 0 and 1 asymptotically follow a two-dimensional Gaussian law via multivariate martingale methods. The rate of convergence is derived analytically. We also study the exact distribution of the degree of a node, as the circuit ages, via a series of Pólya-Eggenberger urn models with “hiccups” in between. The exact expectation and variance of the degree of nodes are determined by recurrence methods. Phase transitions of these degrees are discussed briefly. This is an extension of the abstract (Zhang 2016).

Suggested Citation

  • Panpan Zhang & Hosam M. Mahmoud, 2020. "On Nodes of Small Degrees and Degree Profile in Preferential Dynamic Attachment Circuits," Methodology and Computing in Applied Probability, Springer, vol. 22(2), pages 625-645, June.
    • Handle: RePEc:spr:metcap:v:22:y:2020:i:2:d:10.1007_s11009-019-09726-4
    DOI: 10.1007/s11009-019-09726-4
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    References listed on IDEAS

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    1. Hosam M. Mahmoud, 2014. "The Degree Profile in Some Classes of Random Graphs that Generalize Recursive Trees," Methodology and Computing in Applied Probability, Springer, vol. 16(3), pages 527-538, September.
    2. Mohan Gopaladesikan & Hosam Mahmoud & Mark Daniel Ward, 2014. "Asymptotic Joint Normality of Counts of Uncorrelated Motifs in Recursive Trees," Methodology and Computing in Applied Probability, Springer, vol. 16(4), pages 863-884, December.
    3. José Moler & Fernando Plo & Henar Urmeneta, 2013. "A generalized Pólya urn and limit laws for the number of outputs in a family of random circuits," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(1), pages 46-61, March.
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    Cited by:

    1. Tiandong Wang & Panpan Zhang, 2022. "Directed hybrid random networks mixing preferential attachment with uniform attachment mechanisms," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 74(5), pages 957-986, October.
    2. Hosam Mahmoud, 2022. "Profile of Random Exponential Recursive Trees," Methodology and Computing in Applied Probability, Springer, vol. 24(1), pages 259-275, March.
    3. Raheel Anwar & Muhammad Irfan Yousuf & Muhammad Abid, 2022. "Uniform Preferential Selection Model for Generating Scale-free Networks," Methodology and Computing in Applied Probability, Springer, vol. 24(1), pages 449-470, March.
    4. Feng, Yarong & Mahmoud, Hosam M., 2021. "Dynamic Pólya–Eggenberger urns," Statistics & Probability Letters, Elsevier, vol. 174(C).

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