The winner takes it all

We study competing ¿rst passage percolation on graphs generated by the con¿guration 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 ¿_1 (¿_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 t ¿ (2,3), then, as the number of vertices tends to in¿nity, one of the infection types will almost surely occupy all but a ¿nite 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 ¿_1 and ¿_2. The picture is similar with multiple starting points for the infections.