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We study the growth of two competing infection types on graphs generated bythe configuration model with a given degree sequence. Starting from twovertices chosen uniformly at random, the infection types spread via the edgesin 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 finitesecond moment, we show that if $\lambda_1=\lambda_2$, then the fraction ofvertices that are ultimately infected by type 1 converges to a continuousrandom variable $V\in(0,1)$, as the number of vertices tends to infinity. Bothinfection 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 largerintensity occupies all but a vanishing fraction of the vertices. Our resultsapply also to a uniformly chosen simple graph with the given degree sequence.

Competing first passage percolation on random graphs with finite variance degrees