The Moran process on a random graph

TL;DR

This paper analyzes the fixation probability of mutants spreading on a random graph using the Moran process, providing asymptotic estimates based on initial mutant position and network structure.

Contribution

It offers the first asymptotic estimates of fixation probability for the Moran process on $G_{n,p}$ at the connectivity threshold, considering initial mutant location.

Findings

Fixation probability depends on initial mutant degree and neighbors.

Asymptotic estimates are derived for both Birth-Death and Death-Birth processes.

Results highlight the influence of network structure on mutant spread.

Abstract

We study the fixation probability for two versions of the Moran process on the random graph $G_{n,p}$ at the threshold for connectivity. The Moran process models the spread of a mutant population in a network. Throughtout the process there are vertices of two types, mutants and non-mutants. Mutants have fitness $s$ and non-mutants have fitness 1. The process starts with a unique individual mutant located at the vertex $v_0$. In the Birth-Death version of the process a random vertex is chosen proportional to its fitness and then changes the type of a random neighbor to its own. The process continues until the set of mutants $X$ is empty or $[n]$. In the Death-Birth version a uniform random vertex is chosen and then takes the type of a random neighbor, chosen according to fitness. The process again continues until the set of mutants $X$ is empty or $[n]$. The {\em fixation probability} is the probability that the process ends with $X=\emptyset$. We give asymptotically correct estimates of the fixation probability that depend on degree of $v_0$ and its neighbors.,