Continuous approximations for the fixation probability of the Moran processes on star graphs

TL;DR

This paper develops continuous approximations for the fixation probability of Moran processes on star graphs with frequency-dependent fitness functions, providing accurate solutions and identifying structural effects in evolutionary dynamics.

Contribution

It introduces ODE-based approximations for fixation probabilities under general fitness functions in star graphs, extending previous models to more complex, frequency-dependent scenarios.

Findings

Approximate fixation probabilities have error of order 1/N for death-birth processes.

Star graphs can act as amplifiers, suppressors, or be isothermal depending on initial mutant placement.

Approximations remain accurate for moderate population sizes and diverse fitness functions.

Abstract

We consider a generalized version of the birth-death (BD) and death-birth (DB) processes introduced by Kaveh, Komarova, and Kohandel (2015), in which two constant fitnesses, one for birth and the other for death, describe the selection mechanism of the population. Rather than constant fitnesses, in this paper we consider more general frequency-dependent fitness functions (allowing any smooth functions) under the weak-selection regime. A particular case arises in evolutionary games on graphs, where the fitness functions are linear combinations of the frequencies of types. For a large population structured as a star graph, we provide approximations for the fixation probability which are solutions of certain ODEs (or systems of ODEs). For the DB case, we prove that our approximation has an error of order $1/N$, where $N$ is the size of the population. The general BD and DB processes contain, as special cases, the BD-* and DB-* (where * can be either B or D) processes described in Hadjichrysanthou, Broom, and Rycht\'a\v{r} (2011) -- this class includes many examples of update rules used in the literature. Our analysis shows how the star graph may act as an amplifier, suppressor, or remains isothermal depending on the scaling of the initial mutant placement. We identify an analytical threshold for this transition and illustrate it through applications to evolutionary games, which further highlight asymmetric structural effects across different game types. Numerical examples show that our fixation probability approximations remain accurate even for moderate population sizes and across a wide range of frequency-dependent fitness functions, extending well beyond previously studied linear cases derived from evolutionary games, or constant fitness scenarios.