  The dynamical mechanisms that can stabilize the coexistence of species (or
strategies) are of substantial interest for the maintenance of biodiversity and
in sociobehavioural dynamics. We investigate the mean extinction time in the
coevolutionary dynamics of three cyclically invading strategies for different
evolutionary processes on various classes of complex networks, including random
graphs, scale-free and small world networks. We find that scale-free and random
graphs lead to a strong stabilization of coexistence both for the Moran process
and the Local Update process. The stabilization is of an order of magnitude
stronger compared to a lattice topology, and is mainly caused by the degree
heterogeneity of the graph. However, evolutionary processes on graphs can be
defined in many variants, and we show that in a process using effective payoffs
the effect of the network topology can be completely reversed. Thus,
stabilization of coexistence depends on both network geometry and underlying
evolutionary process.
