When the mean is not enough: Calculating fixation time distributions in birth-death processes

TL;DR

This paper develops methods to compute the full distribution of fixation times in birth-death processes, revealing insights beyond average times, with applications in evolutionary game dynamics.

Contribution

It introduces spectral methods to efficiently calculate fixation time distributions in birth-death processes, extending analysis beyond mean fixation times.

Findings

Distribution of fixation times can be expressed via process spectrum.

Median fixation time may serve as an analog of mixing times.

Efficient sampling methods for fixation time distributions are provided.

Abstract

Studies of fixation dynamics in Markov processes predominantly focus on the mean time to absorption. This may be inadequate if the distribution is broad and skewed. We compute the distribution of fixation times in one-step birth-death processes with two absorbing states. These are expressed in terms of the spectrum of the process, and we provide different representations as forward-only processes in eigenspace. These allow efficient sampling of fixation time distributions. As an application we study evolutionary game dynamics, where invading mutants can reach fixation or go extinct. We also highlight the median fixation time as a possible analog of mixing times in systems with small mutation rates and no absorbing states, whereas the mean fixation time has no such interpretation.