Graphon-valued stochastic processes from population genetics

The Annals of Applied Probability, vol. 31

Abstract

The goal of this paper is to construct a natural class of graphon-valued processes arising from population genetics. We consider finite populations where individuals carry one of finitely many genetic types and change type according to Fisher–Wright 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 types of the two individuals and 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 Fisher–Wright 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.

Authors 3

1. Siva Athreya

   Indian Statistical Institute, Bangalore Centre

   Affiliation as printed

   Theoretical Statistics and Mathematics Division, Indian Statistical Institute, Bangalore Centre

2. Frank den Hollander Aachen Mathematical Institute

   Leiden University

   Affiliation as printed

   Mathematical Institute, Leiden University

3. Adrian Röllin

   National University of Singapore

   Affiliation as printed

   Department of Statistics and Applied Probability, National University of Singapore