Fixation probabilities on superstars, revisited and revised

Population structures can be crucial determinants of evolutionary processes. For the Moran process on graphs certain structures suppress selective pressure, while others amplify it (Lieberman et al. 2005 Nature 433 312-316). Evolutionary amplifiers suppress random drift and enhance selection. Recently, some results for the most powerful known evolutionary amplifier, the superstar, have been invalidated by a counter example (D\'iaz et al. 2013 Proc. R. Soc. A 469 20130193). Here we correct the original proof and derive improved upper and lower bounds, which indicate that the fixation probability remains close to $1-1/(r^4 H)$ for population size $N\to\infty$ and structural parameter $H\gg1$. This correction resolves the differences between the two aforementioned papers. We also confirm that in the limit $N,H\to\infty$ superstars remain capable of eliminating random drift and hence of providing arbitrarily strong selective advantages to any beneficial mutation. In addition, we investigate the robustness of amplification in superstars and find that it appears to be a fragile phenomenon with respect to changes in the selection or mutation processes.