  Evolutionary dynamics has been classically studied for homogeneous
populations, but now there is a growing interest in the non-homogenous case.
One of the most important models has been proposed by Lieberman, Hauert and
Nowak, adapting to a weighted directed graph the classical process described by
Moran. The Markov chain associated with the graph can be modified by erasing
all non-trivial loops in its state space, obtaining the so-called Embedded
Markov chain (EMC). The fixation probability remains unchanged, but the
expected time to absorption (fixation or extinction) is reduced. In this paper,
we shall use this idea to compute asymptotically the average fixation
probability for complete bipartite graphs. To this end, we firstly review some
recent results on evolutionary dynamics on graphs trying to clarify some
points. We also revisit the 'Star Theorem' proved by Lieberman, Hauert and
Nowak for the star graphs. Theoretically, EMC techniques allow fast computation
of the fixation probability, but in practice this is not always true. Thus, in
the last part of the paper, we compare this algorithm with the standard Monte
Carlo method for some kind of complex networks.
