# Methods for approximating stochastic evolutionary dynamics on graphs

## TL;DR

This paper introduces node-level approximation methods for stochastic evolutionary dynamics on complex graphs, enabling more accurate predictions of mutant fixation probabilities compared to traditional models.

## Contribution

It develops novel node-level approximation techniques adapted from statistical physics to analyze complex structured populations in evolutionary graph theory.

## Key findings

- Accurately predicts fixation probabilities on various graphs.
- Effective in capturing individual node dynamics.
- Outperforms existing homogenised models.

## Abstract

Population structure can have a significant effect on evolution. For some systems with sufficient symmetry, analytic results can be derived within the mathematical framework of evolutionary graph theory which relate to the outcome of the evolutionary process. However, for more complicated heterogeneous structures, computationally intensive methods are required such as individual-based stochastic simulations. By adapting methods from statistical physics, including moment closure techniques, we first show how to derive existing homogenised pair approximation models and the exact neutral drift model. We then develop node-level approximations to stochastic evolutionary processes on arbitrarily complex structured populations represented by finite graphs, which can capture the different dynamics for individual nodes in the population. Using these approximations, we evaluate the fixation probability of invading mutants for given initial conditions, where the dynamics follow standard evolutionary processes such as the invasion process. Comparisons with the output of stochastic simulations reveal the effectiveness of our approximations in describing the stochastic processes and in predicting the probability of fixation of mutants on a wide range of graphs. Construction of these models facilitates a systematic analysis and is valuable for a greater understanding of the influence of population structure on evolutionary processes.