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Degree growth rates and index estimation in a directed preferential attachment model

Author

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  • Wang, Tiandong
  • Resnick, Sidney I.

Abstract

Preferential attachment is widely used to model power-law behavior of degree distributions in both directed and undirected networks. In a directed preferential attachment model, despite the well-known marginal power-law degree distributions, not much investigation has been done on the joint behavior of the in- and out-degree growth. Also, statistical estimates of the marginal tail exponent of the power-law degree distribution often use the Hill estimator as one of the key summary statistics, even though no theoretical justification has been given. This paper focuses on the convergence of the joint empirical measure for in- and out-degrees and proves the consistency of the Hill estimator. To do this, we first derive the asymptotic behavior of the joint degree sequences by embedding the in- and out-degrees of a fixed node into a pair of switched birth processes with immigration and then establish the convergence of the joint tail empirical measure. From these steps, the consistency of the Hill estimators is obtained.

Suggested Citation

  • Wang, Tiandong & Resnick, Sidney I., 2020. "Degree growth rates and index estimation in a directed preferential attachment model," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 878-906.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:2:p:878-906
    DOI: 10.1016/j.spa.2019.03.021
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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. Natalia Markovich & Marijus Vaičiulis, 2023. "Extreme Value Statistics for Evolving Random Networks," Mathematics, MDPI, vol. 11(9), pages 1-35, May.
    3. Tiandong Wang & Sidney Resnick, 2023. "Poisson Edge Growth and Preferential Attachment Networks," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-25, March.
    4. Natalia Markovich & Maksim Ryzhov & Marijus Vaičiulis, 2022. "Tail Index Estimation of PageRanks in Evolving Random Graphs," Mathematics, MDPI, vol. 10(16), pages 1-26, August.

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