# The winner takes it all

### Uittreksel

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.
Originele taal-2 Engels Eindhoven Eurandom 22 Gepubliceerd - 2013

### Publicatie series

Naam Report Eurandom 2013024 1389-2355

### Vingerafdruk

Infection
Vertex of a graph
Degree Distribution
Power Law
Exponent
Tend
Graph in graph theory
Model

### Citeer dit

Deijfen, M., & Hofstad, van der, R. W. (2013). The winner takes it all. (Report Eurandom; Vol. 2013024). Eindhoven: Eurandom.
Deijfen, M. ; Hofstad, van der, R.W. / The winner takes it all. Eindhoven : Eurandom, 2013. 22 blz. (Report Eurandom).
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title = "The winner takes it all",
abstract = "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.",
author = "M. Deijfen and {Hofstad, van der}, R.W.",
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Deijfen, M & Hofstad, van der, RW 2013, The winner takes it all. Report Eurandom, vol. 2013024, Eurandom, Eindhoven.

The winner takes it all. / Deijfen, M.; Hofstad, van der, R.W.

Eindhoven : Eurandom, 2013. 22 blz. (Report Eurandom; Vol. 2013024).

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AB - 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.

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Deijfen M, Hofstad, van der RW. The winner takes it all. Eindhoven: Eurandom, 2013. 22 blz. (Report Eurandom).