Fixation times on directed graphs

TL;DR

This paper investigates fixation times of mutants on directed graphs, proving conditions for rapid fixation, providing an algorithm for upper bounds, and identifying graph classes with universally fast fixation times.

Contribution

It introduces new theoretical results on fixation times on directed graphs, including conditions for speed, an algorithm for bounds, and characterization of graphs with fast fixation.

Findings

Fixation time is fast for any directed graph if the mutant's advantage is large enough.

An efficient algorithm provides upper bounds for fixation times across all graphs and advantages.

Certain classes of directed graphs, including known amplifiers, have universally fast fixation times.

Abstract

Computing the rate of evolution in spatially structured populations is difficult. A key quantity is the fixation time of a single mutant with relative reproduction rate $r$ which invades a population of residents. We say that the fixation time is "fast" if it is at most a polynomial function in terms of the population size $N$. Here we study fixation times of advantageous mutants ($r>1$) and neutral mutants ($r=1$) on directed graphs, which are those graphs that have at least some one-way connections. We obtain three main results. First, we prove that for any directed graph the fixation time is fast, provided that $r$ is sufficiently large. Second, we construct an efficient algorithm that gives an upper bound for the fixation time for any graph and any $r\ge 1$. Third, we identify a broad class of directed graphs with fast fixation times for any $r\ge 1$. This class includes previously studied amplifiers of selection, such as Superstars and Metafunnels. We also show that on some graphs the fixation time is not a monotonically declining function of $r$; in particular, neutral fixation can occur faster than fixation for small selective advantages.