On the Fixation Probability of Superstars

TL;DR

This paper critically examines the fixation probability of superstars in graph-structured populations, correcting a previous claim and providing new bounds based on larger simulations, challenging earlier assumptions about their effectiveness.

Contribution

The paper refutes the original claim that fixation probability tends to 1-r^{-k} for superstars, offering a corrected upper bound and extensive simulations to support this.

Findings

The fixation probability for k=5 is at most 1-1/j(r), with j(r)=Θ(r^4).

Simulations on larger graphs do not support the original claim of fixation probability approaching 1 as k increases.

The qualitative trend that fixation probability increases with k remains plausible, but the quantitative bounds differ from prior claims.

Abstract

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.