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Tail Index Estimation of PageRanks in Evolving Random Graphs

Author

Listed:
  • Natalia Markovich

    (V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, 117997 Moscow, Russia)

  • Maksim Ryzhov

    (V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, 117997 Moscow, Russia)

  • Marijus Vaičiulis

    (Institute of Data Science and Digital Technologies, Vilnius University, Akademijos St. 4, LT-08663 Vilnius, Lithuania)

Abstract

Random graphs are subject to the heterogeneities of the distributions of node indices and their dependence structures. Superstar nodes to which a large proportion of nodes attach in the evolving graphs are considered. In the present paper, a statistical analysis of the extremal part of random graphs is considered. We used the extreme value theory regarding sums and maxima of non-stationary random length sequences to evaluate the tail index of the PageRanks and max-linear models of superstar nodes in the evolving graphs where existing nodes or edges can be deleted or not. The evolution is provided by a linear preferential attachment. Our approach is based on the analysis of maxima and sums of the node PageRanks over communities (block maxima and block sums), which can be independent or weakly dependent random variables. By an empirical study, it was found that tail indices of the block maxima and block sums are close to the minimum tail index of representative series extracted from the communities. The tail indices are estimated by data of simulated graphs.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:16:p:3026-:d:894730
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    References listed on IDEAS

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    1. Attila Mester & Andrei Pop & Bogdan-Eduard-Mădălin Mursa & Horea Greblă & Laura Dioşan & Camelia Chira, 2021. "Network Analysis Based on Important Node Selection and Community Detection," Mathematics, MDPI, vol. 9(18), pages 1-16, September.
    2. Danielsson, Jon & Ergun, Lerby M. & Haan, Laurens de & Vries, Casper G. de, 2016. "Tail index estimation: quantile driven threshold selection," LSE Research Online Documents on Economics 66193, London School of Economics and Political Science, LSE Library.
    3. 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.
    4. Hall, Peter, 1990. "Using the bootstrap to estimate mean squared error and select smoothing parameter in nonparametric problems," Journal of Multivariate Analysis, Elsevier, vol. 32(2), pages 177-203, February.
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    Cited by:

    1. Natalia Markovich, 2024. "Extremal properties of evolving networks: local dependence and heavy tails," Annals of Operations Research, Springer, vol. 339(3), pages 1839-1870, August.
    2. Natalia Markovich & Marijus Vaičiulis, 2023. "Extreme Value Statistics for Evolving Random Networks," Mathematics, MDPI, vol. 11(9), pages 1-35, May.

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