The Annals of Applied Probability

The winner takes it all

Maria Deijfen and Remco van der Hofstad

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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

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

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

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Zentralblatt MATH identifier

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

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


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.

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