The grapheme-valued Wright–Fisher diffusion with mutation

Theoretical Population Biology, vol. 158, pp. 76–88

Abstract

In Athreya et al. (2021), models from population genetics were used to define stochastic dynamics in the space of graphons arising as continuum limits of dense graphs. In the present paper we exhibit an example of a simple neutral population genetics model for which this dynamics is a Markovian diffusion that can be characterized as the solution of a martingale problem. In particular, we consider a Markov chain in the space of finite graphs that resembles a Moran model with resampling and mutation. We encode the finite graphs as graphemes, which can be represented as a triple consisting of a vertex set (or more generally, a topological space), an adjacency matrix, and a sampling (Borel) measure. We equip the space of graphons with convergence of sample subgraph densities and show that the grapheme-valued Markov chain converges to a grapheme-valued diffusion as the number of vertices goes to infinity. We show that the grapheme-valued diffusion has a stationary distribution that is linked to the Griffiths-Engen-McCloskey (GEM) distribution. In a companion paper (Greven et al. 2023), we build up a general theory for obtaining grapheme-valued diffusions via genealogies of models in population genetics.

Authors 4

1. Andreas Greven

   Friedrich-Alexander-Universität Erlangen-Nürnberg

   Affiliation as printed

   Department Mathematik, Universität Erlangen-Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany. Electronic address: greven@mi.uni-erlangen.de

2. Frank den Hollander Aachen Mathematisch Instituut

   Leiden University

   Affiliation as printed

   Mathematisch Instituut, Universiteit Leiden, Niels Bohrweg 1, 2333 CA, Leiden, The Netherlands. Electronic address: denholla@math.leidenuniv.nl

3. Anton Klimovsky

   University of Würzburg

   Affiliation as printed

   Institut für Mathematik, Emil-Fischer-Strasse 30, 97074 Würzburg, Germany. Electronic address: anton.klimovsky@mathematik.uni-wuerzburg.de

4. Anita Winter

   University of Duisburg-Essen

   Affiliation as printed

   Fakultät für Mathematik, Universität Duisburg-Essen, Campus Essen, Universitätsstraße 2, 45132 Essen, Germany. Electronic address: anita.winter@uni-due.de