  We study the growth of two competing infection types on graphs generated by
the configuration model with a given degree sequence. Starting from two
vertices chosen uniformly at random, the infection types spread via the edges
in the graph in that an uninfected vertex becomes type 1 (2) infected at rate
$\lambda_1$ ($\lambda_2$) times the number of nearest neighbors of type 1 (2).
Assuming (essentially) that the degree of a randomly chosen vertex has finite
second moment, we show that if $\lambda_1=\lambda_2$, then the fraction of
vertices that are ultimately infected by type 1 converges to a continuous
random variable $V\in(0,1)$, as the number of vertices tends to infinity. Both
infection types hence occupy a positive (random) fraction of the vertices. If
$\lambda_1\neq \lambda_2$, on the other hand, then the type with the larger
intensity occupies all but a vanishing fraction of the vertices. Our results
apply also to a uniformly chosen simple graph with the given degree sequence.
