  Evolutionary dynamics on graphs can lead to many interesting and
counterintuitive findings. We study the Moran process, a discrete time
birth-death process, that describes the invasion of a mutant type into a
population of wild-type individuals. Remarkably, the fixation probability of a
single mutant is the same on all regular networks. But non-regular networks can
increase or decrease the fixation probability. While the time until fixation
formally depends on the same transition probabilities as the fixation
probabilities, there is no obvious relation between them. For example, an
amplifier of selection, which increases the fixation probability and thus
decreases the number of mutations needed until one of them is successful, can
at the same time slow down the process of fixation. Based on small networks, we
show analytically that (i) the time to fixation can decrease when links are
removed from the network and (ii) the node providing the best starting
conditions in terms of the shortest fixation time depends on the fitness of the
mutant. Our results are obtained analytically on small networks, but numerical
simulations show that they are qualitatively valid even in much larger
populations.
