# Fitness dependence of the fixation-time distribution for evolutionary dynamics on graphs

## TL;DR

This paper analyzes how mutant fitness influences the distribution of fixation times in evolutionary graph models, revealing distinct behaviors on different network structures and identifying conditions for universal and skewed distributions.

## Contribution

It provides exact solutions for fixation-time distributions under varying fitness levels on complete graphs and rings, highlighting their dependence and discontinuities.

## Key findings

- Fixation times are normally distributed on rings with non-neutral fitness.
- On complete graphs, fixation times follow a fitness-weighted convolution of Gumbel distributions.
- Neutral fitness leads to highly skewed fixation-time distributions on both networks.

## Abstract

Evolutionary graph theory models the effects of natural selection and random drift on structured populations of competing mutant and non-mutant individuals. Recent studies have found that fixation times in such systems often have right-skewed distributions. Little is known, however, about how these distributions and their skew depend on mutant fitness. Here we calculate the fitness dependence of the fixation-time distribution for the Moran Birth-death process in populations modeled by two extreme networks: the complete graph and the one-dimensional ring lattice, obtaining exact solutions in the limit of large network size. We find that with non-neutral fitness, the Moran process on the ring has normally distributed fixation times, independent of the relative fitness of mutants and non-mutants. In contrast, on the complete graph, the fixation-time distribution is a fitness-weighted convolution of two Gumbel distributions. When fitness is neutral the fixation-time distribution jumps discontinuously and becomes highly skewed on both the complete graph and the ring. Even on these simple networks, the fixation-time distribution exhibits rich fitness dependence, with discontinuities and regions of universality. Extensions of our results to two-fitness Moran models, times to partial fixation, and evolution on random networks are discussed.