Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: nlin
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-ex
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The categories of this paper are: acc-phys
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The categories of this paper are: q-alg
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The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-th
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The categories of this paper are: cond-mat
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-fin
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The categories of this paper are: econ
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The categories of this paper are: eess
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The categories of this paper are: hep-ph
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The categories of this paper are: stat