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Estimation of the Tail Index of PageRanks in Random Graphs

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Distributed Computer and Communication Networks: Control, Computation, Communications (DCCN 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13766 ))

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Abstract

Superstar nodes to which a large proportion of nodes attach in the evolving graphs are considered. We attract results of the extreme value theory regarding sums and maxima of non-stationary random length sequences to predict the tail index of the PageRanks and Max-linear models as influence measures of superstar nodes. To this end, the graphs are divided into mutually weakly dependent communities. Maxima and sums of the PageRanks over communities are used as weakly independent block-data. Tail indices of the block-maxima and block-sums and hence, of the PageRanks and the Max-linear models are found to be close to the minimum tail index of series of representative nodes taken from the communities. The graph evolution is provided by a linear preferential attachment. The tail indices are estimated by data of simulated and real temporal graphs.

The reported study was funded by the Russian Science Foundation RSF, project number 22-21-00177 (recipient N.M. Markovich, conceptualization, methodology development, formal analysis, writing–original draft preparation; recipient M.S. Ryzhov, software, data validation).

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Notes

  1. 1.

    The classification into the classes of communities is caused by the Loivan’s algorithm. The classification is not so explicit for evolving simulated graphs in Fig. 2 that might be due to stationary distributions of their in- and out-degrees.

References

  1. Volkovich, Y.V., Litvak, N.: Asymptotic analysis for personalized web search. Adv. Appl. Prob. 42(2), 577–604 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Jelenkovic, P.R., Olvera-Cravioto, M.: Information ranking and power laws on trees. Adv. Appl. Prob. 42(4), 1057–1093 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Markovich, N.M., Ryzhov, M., Krieger, U.R.: Nonparametric analysis of extremes on web graphs: PageRank versus max-linear model. Commun. Comput. Inf. Sci. 700, 13–26 (2017)

    MATH  Google Scholar 

  4. Dugué, N., Perez, A.: Directed Louvain : maximizing modularity in directed networks // [Research Report] Université d’Orléans, hal-01231784 (2015)

    Google Scholar 

  5. Fortunato, S.: Community detection in graphs. Phys. Rep. 486(3), 75–174 (2010)

    Article  MathSciNet  Google Scholar 

  6. Mester, A., Pop, A., Mursa, B.-E.-M., Grebla, H., Diosan, L., Chira, C.: Network analysis based on important node selection and community detection. Mathematics 9, 2294 (2021)

    Article  Google Scholar 

  7. Wan, P., Wang, T., Davis, R.A., Resnick, S.I.: Are extreme value estimation methods useful for network data? Extremes. 23, 171–195 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  8. Das, B., Resnick, S.I.: QQ plots, random sets and data from a heavy tailed distribution. Stoch. Model. 24(1), 103–132 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Markovich, N., Rodionov, I.: Maxima and sums of non-stationary random length sequences. Extremes 23(9), 451–464 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  10. Markovich, N.: Extremes of Sums and Maxima with Application to Random Networks (2021). arXiv:math.PR/2110.04120

  11. Brin, S., Page, L.: The anatomy of a large-scale hypertextual web search engine. Comput. Netw. ISDN Syst. 30, 107–117 (1998)

    Article  Google Scholar 

  12. Markovich, N.: Nonparametric Analysis of Univariate Heavy-Tailed Data: Research and Practice. John Wiley & Sons, New York (2007)

    Google Scholar 

  13. Michail, O.: An introduction to temporal graphs: an algorithmic perspective. In: Zaroliagis, C., Pantziou, G., Kontogiannis, S. (eds.) Algorithms, Probability, Networks, and Games. LNCS, vol. 9295, pp. 308–343. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-24024-4_18

    Chapter  Google Scholar 

  14. Leskovec, J., Krevl, A.: SNAP Datasets: Stanford Large Network Dataset Collection (2014).http://snap.stanford.edu/data/

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Author information

Authors and Affiliations

  1. V.A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Profsoyuznaya Str. 65, 117997, Moscow, Russia

    Natalia M. Markovich & Maksim S. Ryzhov

Authors
  1. Natalia M. Markovich
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  2. Maksim S. Ryzhov
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Corresponding author

Correspondence to Natalia M. Markovich .

Editor information

Editors and Affiliations

  1. V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia

    Vladimir M. Vishnevskiy

  2. Peoples' Friendship University of Russia, Moscow, Russia

    Konstantin E. Samouylov

  3. V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia

    Dmitry V. Kozyrev

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Markovich, N.M., Ryzhov, M.S. (2022). Estimation of the Tail Index of PageRanks in Random Graphs. In: Vishnevskiy, V.M., Samouylov, K.E., Kozyrev, D.V. (eds) Distributed Computer and Communication Networks: Control, Computation, Communications. DCCN 2022. Lecture Notes in Computer Science, vol 13766 . Springer, Cham. https://doi.org/10.1007/978-3-031-23207-7_7

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  • DOI: https://doi.org/10.1007/978-3-031-23207-7_7

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-23206-0

  • Online ISBN: 978-3-031-23207-7

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