# 19.17 Existence of a universal amplifier of selection

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 $$p_n$$?