Exact numerical calculation of fixation probability and time on graphs

TL;DR

This paper introduces a numerical algorithm to accurately compute fixation probabilities and times in the Moran process on small graphs, overcoming the limitations of simulation-based methods.

Contribution

The authors develop an automated, transition matrix-based algorithm for exact calculation of fixation metrics on arbitrary small graphs, supporting different update mechanisms.

Findings

Algorithm provides exact fixation probabilities and times.

Implementation is fast and flexible for various graph types.

Enables interactive analysis of graph effects on evolution.

Abstract

The Moran process on graphs is a popular model to study the dynamics of evolution in a spatially structured population. Exact analytical solutions for the fixation probability and time of a new mutant have been found for only a few classes of graphs so far. Simulations are time-expensive and many realizations are necessary, as the variance of the fixation times is high. We present an algorithm that numerically computes these quantities for arbitrary small graphs by an approach based on the transition matrix. The advantage over simulations is that the calculation has to be executed only once. Building the transition matrix is automated by our algorithm. This enables a fast and interactive study of different graph structures and their effect on fixation probability and time. We provide a fast implementation in C with this note https://github.com/hindersin/efficientFixation. Our code is very flexible, as it can handle two different update mechanisms (Birth-death or death-Birth), as well as arbitrary directed or undirected graphs.