On the Fixation Probability of Superstars

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 describe a class of graphs (called “superstars”), with a parameter $k$. Superstars are designed to have an increasing fixation probability as $k$ increases. They state that the probability of fixation tends to $1-r^{-k}$ as graphs get larger but we show that this claim is untrue as stated. Specifically, for $k=5$, we show that the true fixation probability (in the limit, as graphs get larger) is at most $1-1/j(r)$ where $j(r)=\Theta(r^4)$, contrary to the claimed result. We do believe that the qualitative claim of Lieberman et al.\ — that the fixation probability of superstars tends to 1 as $k$ increases — is correct, and that it can probably be proved along the lines of their sketch. We were able to run larger computer simulations than the ones presented in their paper. However, simulations on graphs of around 40,000 vertices do not support their claim. Perhaps these graphs are too small to exhibit the limiting behaviour.

💡 Research Summary

The paper revisits the celebrated result of Lieberman, Hauert, and Nowak (Nature, 2005) concerning the fixation probability of mutants on “superstar” graphs, a class of directed graphs designed to amplify selection. In the original work the authors claimed that for a superstar with parameter k, the probability that a single mutant of relative fitness r eventually fixes in an infinitely large population converges to (1-r^{-k}). Their argument relied on a heuristic decomposition of the Moran process into two independent stages: (i) the mutant reaches the central hub, and (ii) from the hub it spreads to the whole graph.

The present study focuses on the case k = 5 and provides a rigorous counter‑example to the claimed formula. By constructing the exact Markov chain that describes the Moran dynamics on a 5‑level superstar, the authors compute the transition probabilities for a mutant moving along the peripheral chains toward the hub. They show that the probability of successful traversal of a chain is not simply proportional to (r^{5}) as assumed, but is instead limited by a term of order (r^{-4}). Consequently, the fixation probability satisfies the upper bound
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