Local Neighbourhoods for First-Passage Percolation on the Configuration Model

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Abstract

We consider first-passage percolation on the configuration model. Once the network has been generated each edge is assigned an i.i.d. weight modeling the passage time of a message along this edge. Then independently two vertices are chosen uniformly at random, a sender and a recipient, and all edges along the geodesic connecting the two vertices are coloured in red (in the case that both vertices are in the same component). In this article we prove local limit theorems for the coloured graph around the recipient in the spirit of Benjamini and Schramm. We consider the explosive regime, in which case the random distances are of finite order, and the Malthusian regime, in which case the random distances are of logarithmic order.

Keywords

First passage percolation Random graphs Configuration model Local limit Geodesics Branching processes 

Mathematics Subject Classification

Primary 05C80 Secondary 60J80 

References

  1. 1.
    Aldous , D., Steele, J.M.: The objective method: probabilistic combinatorial optimization and local weak convergence. In: Probability on discrete structures, volume 110 of Encyclopaedia Math. Sci., pp. 1–72. Springer, Berlin (2004)Google Scholar
  2. 2.
    Amini, O., Devroye, L., Griffiths, S., Olver, N.: On explosions in heavy-tailed branching random walks. Ann. Probab. 41(3B), 1864–1899 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Auffinger, A., Damron, M., Hanson, J.: 50 years of first passage percolation. To appear in AMS University Lecture Series, arXiv:1511.03262 (2017)
  4. 4.
    Baroni, E., van der Hofstad, R., Komjáthy, J.: Nonuniversality of weighted random graphs with infinite variance degree. J. Appl. Probab. 54(1), 146–164 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Benjamini, I., Schramm, O.: Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6(23), 13 (2001)MathSciNetMATHGoogle Scholar
  6. 6.
    Bhamidi, S.: First passage percolation on locally treelike networks. I. Dense random graphs. J. Math. Phys. 49(12), 125218 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bhamidi, S., Goodman, J., van der Hofstad, R., Komjáthy, J.: Degree distribution of shortest path trees and bias of network sampling algorithms. Ann. Appl. Probab. 25(4), 1780–1826 (2015)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bhamidi, S., van der Hofstad, R., Hooghiemstra, G.: Extreme value theory, Poisson–Dirichlet distributions, and first passage percolation on random networks. Adv. Appl. Probab. 42(3), 706–738 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bhamidi, S., van der Hofstad, R., Hooghiemstra, G.: First passage percolation on the Erdős–Rényi random graph. Comb. Probab. Comput. 20(5), 683–707 (2011)CrossRefMATHGoogle Scholar
  10. 10.
    Bhamidi, S., van der Hofstad, R., Hooghiemstra, G.: First passage percolation on random graphs with finite mean degrees. Ann. Appl. Probab. 20(5), 1907–1965 (2010)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bhamidi, S., van der Hofstad, R., Hooghiemstra, G.: Universality for first passage percolation on sparse random graphs. Ann. Probab. 45(4), 2568–2630 (2017)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Hammersley, J.M., Welsh, D.J.A: First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In Proc. Internat. Res. Semin., Statist. Lab., Univ. California, Berkeley, CA, pp. 61–110. Springer-Verlag, New York (1965)Google Scholar
  13. 13.
    van der Hofstad, R.: Random Graphs and Complex Networks. Cambridge Series in Statistical and Probabilistic Mathematics, vol. 1. Cambridge University Press, Cambridge (2017)CrossRefMATHGoogle Scholar
  14. 14.
    van der Hofstad, R.: Random Graphs and Complex Networks, vol. 2 (2017, in preparation). http://www.win.tue.nl/~rhofstad/
  15. 15.
    van der Hofstad, R.: Stochastic processes on random graphs. In: Lecture Notes for the 47th Summer School in Probability, Saint-Flour (2017)Google Scholar
  16. 16.
    van der Hofstad, R., Hooghiemstra, G., Van Mieghem, P.: First-passage percolation on the random graph. Probab. Eng. Inf. Sci. 15(2), 225–237 (2001)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Jagers, P., Nerman, O.: The growth and composition of branching populations. Adv. Appl. Probab. 16(2), 221–259 (1984)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Komjáthy, J.: Explosive Crump-Mode-Jagers branching processes. Preprint, arXiv:1602.01657 (2016)

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für Mathematische StatistikWestfälische Wilhelms-Universität MünsterMünsterGermany
  2. 2.Department of Mathematical SciencesUniversity of BathBathUK

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