The Annals of Applied Probability

The winner takes it all

Maria Deijfen and Remco van der Hofstad

Full-text: Open access

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.