# Graphon-valued stochastic processes from population genetics

## TL;DR

This paper develops a theory for graphon-valued stochastic processes derived from population genetics, modeling the evolution of large populations with type-dependent connections and analyzing their limits as population size and types grow.

## Contribution

It introduces a new framework for graphon-valued processes in population genetics and characterizes their behavior in large-population and large-type limits.

## Key findings

- Convergence of adjacency matrix processes to graphon-valued processes in large populations.
- Description of the limiting process driven by Wright-Fisher and Fleming-Viot diffusions.
- Extension of the theory to infinite types via a type-connection kernel.

## Abstract

The goal of this paper is to develop a theory of graphon-valued stochastic processes, and to construct and analyse a natural class of such processes arising from population genetics. We consider finite populations where individuals change type according to Wright-Fisher resampling. At any time, each pair of individuals is linked by an edge with a probability that is given by a type-connection matrix, whose entries depend on the current empirical type distribution of the entire population via a fitness function. We show that, in the large-population-size limit and with an appropriate scaling of time, the evolution of the associated adjacency matrix converges to a random process in the space of graphons, driven by the type-connection matrix and the underlying Wright-Fisher diffusion on the multi-type simplex. In the limit as the number of types tends to infinity, the limiting process is driven by the type-connection kernel and the underlying Fleming-Viot diffusion.