The Annals of Applied Probability

The winner takes it all

Maria Deijfen and Remco van der Hofstad

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study competing first passage percolation on graphs generated by the configuration model. At time 0, vertex 1 and vertex 2 are infected with the type 1 and the type 2 infection, respectively, and an uninfected vertex then becomes type 1 (2) infected at rate $\lambda_{1}$ ($\lambda_{2}$) times the number of edges connecting it to a type 1 (2) infected neighbor. Our main result is that, if the degree distribution is a power-law with exponent $\tau\in(2,3)$, then as the number of vertices tends to infinity and with high probability, one of the infection types will occupy all but a finite number of vertices. Furthermore, which one of the infections wins is random and both infections have a positive probability of winning regardless of the values of $\lambda_{1}$ and $\lambda_{2}$. The picture is similar with multiple starting points for the infections.

Article information

Source
Ann. Appl. Probab., Volume 26, Number 4 (2016), 2419-2453.

Dates
Received: June 2013
Revised: April 2015
First available in Project Euclid: 1 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1472745463

Digital Object Identifier
doi:10.1214/15-AAP1151

Mathematical Reviews number (MathSciNet)
MR3543901

Zentralblatt MATH identifier
1352.60129

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 05C80: Random graphs [See also 60B20] 90B15: Network models, stochastic

Keywords
Random graphs configuration model first passage percolation competing growth coexistence continuous-time branching process

Citation

Deijfen, Maria; van der Hofstad, Remco. The winner takes it all. Ann. Appl. Probab. 26 (2016), no. 4, 2419--2453. doi:10.1214/15-AAP1151. https://projecteuclid.org/euclid.aoap/1472745463


Export citation

References

  • [1] Antunovic, T., Dekel, Y., Mossel, E. and Peres, Y. (2011). Competing first passage percolation on random regular graphs. Preprint. Available at arXiv:1109.2575.
  • [2] Antunovic, T., Mossel, E. and Racz, M. (2014). Coexistence in preferential attachment networks. Preprint. Available at arXiv:1307.2893.
  • [3] Baroni, E., van der Hofstad, R. and Komjáthy, J. (2014). Fixed speed competition on the configuration model with infinite variance degrees: Unequal speeds. Preprint. Available at arXiv:1408.0475.
  • [4] Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2010). First passage percolation on random graphs with finite mean degrees. Ann. Appl. Probab. 20 1907–1965.
  • [5] Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2011). First passage percolation on the Erdős–Rényi random graph. Combin. Probab. Comput. 20 683–707.
  • [6] Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2012). Universality for first passage percolation on sparse random graph. Preprint. Available at arXiv:1210.6839.
  • [7] Cox, J. T. and Durrett, R. (1981). Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probab. 9 583–603.
  • [8] Garet, O. and Marchand, R. (2005). Coexistence in two-type first-passage percolation models. Ann. Appl. Probab. 15 298–330.
  • [9] Grey, D. R. (1973/74). Explosiveness of age-dependent branching processes. Z. Wahrsch. Verw. Gebiete 28 129–137.
  • [10] Grimmett, G. and Kesten, H. (1984). First-passage percolation, network flows and electrical resistances. Z. Wahrsch. Verw. Gebiete 66 335–366.
  • [11] Häggström, O. and Pemantle, R. (1998). First passage percolation and a model for competing spatial growth. J. Appl. Probab. 35 683–692.
  • [12] Häggström, O. and Pemantle, R. (2000). Absence of mutual unbounded growth for almost all parameter values in the two-type Richardson model. Stochastic Process. Appl. 90 207–222.
  • [13] Hammersley, J. M. and Welsh, D. J. A. (1965). First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In Proc. Internat. Res. Semin., Statist. Lab., Univ. California, Berkeley, Calif 61–110. Springer, New York.
  • [14] Hoffman, C. (2005). Coexistence for Richardson type competing spatial growth models. Ann. Appl. Probab. 15 739–747.
  • [15] Janson, S. (2009). The probability that a random multigraph is simple. Combin. Probab. Comput. 18 205–225.
  • [16] Janson, S. and Luczak, M. J. (2009). A new approach to the giant component problem. Random Structures Algorithms 34 197–216.
  • [17] Kesten, H. (1993). On the speed of convergence in first-passage percolation. Ann. Appl. Probab. 3 296–338.
  • [18] Lyons, R. and Pemantle, R. (1992). Random walk in a random environment and first-passage percolation on trees. Ann. Probab. 20 125–136.
  • [19] Molloy, M. and Reed, B. (1995). A critical point for random graphs with a given degree sequence. In Proceedings of the Sixth International Seminar on Random Graphs and Probabilistic Methods in Combinatorics and Computer Science, “Random Graphs ’93 (Poznań, 1993) 6 161–179.
  • [20] Molloy, M. and Reed, B. (1998). The size of the giant component of a random graph with a given degree sequence. Combin. Probab. Comput. 7 295–305.
  • [21] Newman, C. M. and Piza, M. S. T. (1995). Divergence of shape fluctuations in two dimensions. Ann. Probab. 23 977–1005.
  • [22] Richardson, D. (1973). Random growth in a tessellation. Proc. Cambridge Philos. Soc. 74 515–528.
  • [23] Smythe, R. T. and Wierman, J. C. (1978). First-Passage Percolation on the Square Lattice. Lecture Notes in Math. 671. Springer, Berlin.
  • [24] van der Hofstad, R. (2013). Random graphs and complex networks. Available at www.win.tue.nl/~rhofstad.
  • [25] van der Hofstad, R., Hooghiemstra, G. and Znamenski, D. (2007). A phase transition for the diameter of the configuration model. Internet Math. 4 113–128.
  • [26] van der Hofstad, R. and Komjáthy, J. Fixed speed competition on the configuration model with infinite variance degrees: Equal speeds. Unpublished manuscript.