Moran Birth-death processes are stochastic processes defined as follows.
Consider a connected graph \(G\) and a parameter
\(r \in \mathbb{R}\) strictly greater than \(1\).
We start with a partition of the vertices of \(G\) into two types:
*mutant* vertices, that have *fitness* \(r\),
and *resident* vertices, that have fitness \(1\).
At each step, a random vertex is chosen
with probability proportional to its fitness,
and spreads its type to an adjacent vertex
chosen uniformly at random.

With probability \(1\),
the process eventually reaches
either fixation of the mutation (all the vertices are mutant)
or extinction (all the vertices are resident).
The *fixation probability* of a vertex \(v\) of \(G\)
is the probability that the process starting with
a single mutant at \(v\) eventually reaches fixation.
The fixation probability of each vertex of the complete graph on \(n\in \mathbb{N}\) vertices is
\[
p_n = \frac{1-r^{-1}}{1-r^{-n}}.
\]

**Open problem:**
Does there exist a connected graph
\(G\) with \(n \in \mathbb{N}\) vertices such that
the fixation probability of each vertex of \(G\)
is strictly greater than \(p_n\)?