Can Fixation be Guaranteed in the Generalized Moran Process?

The Moran process models the spread of genetic mutations through a population. A mutant with relative fitness r is introduced into a population and the system evolves, either reaching fixation (in which every individual is a mutant) or extinction (in which none is). In a widely cited paper (Nature, 2005), Lieberman, Hauert and Nowak generalize the model to populations on the vertices of graphs. They claim to have discovered a class of graphs (called “superstars”), with a parameter k, within which the probability of fixation tends to 1− r−k as graphs get larger. Thus, they say that these graphs “guarantee the fixation of any advantageous mutant”. They give a non-rigorous proof of the claimed limiting fixation probability, which we show to be incorrect. Specifically, for k = 5, we show that the true fixation probability is at most 1 − 1/j(r) where j(r) = Θ(r), contrary to the claimed result. Since the claimed proof is flawed, and no others are known, we investigate the claim via simulation. Lieberman et al. verified their claim for certain small values of r and k by simulation. We performed simulations over a wider range of parameters. These show, with 99.5% confidence, that the formula also does not apply for these other values of k. It is hard to draw conclusions about limiting behaviour from simulations but the simulations do not seem to support the claim that, for fixed r > 1, the fixation probability tends to 1 as k increases. It is an interesting open question whether the class of graphs introduced by Lieberman et al. have this property and, indeed, whether any class does.