The winner takes it all

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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 λ1 (λ2 ) 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 τ ⋯ (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 λ1 and λ2 . The picture is similar with multiple starting points for the infections.

Original languageEnglish
Pages (from-to)2419-2453
Number of pages35
JournalThe Annals of Applied Probability
Issue number4
Publication statusPublished - 1 Aug 2016


  • Coexistence
  • Competing growth
  • Configuration model
  • Continuous-time branching process.
  • First passage percolation
  • Random graphs


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