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The grapheme-valued Wright-Fisher diffusion with mutation.

Andreas Greven1, Frank den Hollander2, Anton Klimovsky3

  • 1Department Mathematik, Universität Erlangen-Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany.

Theoretical Population Biology
|May 31, 2024
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Summary
This summary is machine-generated.

This study introduces a neutral population genetics model exhibiting Markovian diffusion dynamics. The grapheme-valued Markov chain converges to a diffusion, revealing a stationary distribution linked to the Griffiths-Engen-McCloskey (GEM) distribution.

Keywords:
Ewens sampling formulaGEM-distributionGraph-valued Markov chainGraphonsInfinite alleles modelPopulation genetics

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Area of Science:

  • Mathematical Biology
  • Population Genetics
  • Stochastic Processes

Background:

  • Previous work (Athreya et al., 2021) defined stochastic dynamics on graphons using population genetics models.
  • Graphons represent continuum limits of dense graphs, providing a framework for large-scale network analysis.

Purpose of the Study:

  • To present a neutral population genetics model demonstrating Markovian diffusion dynamics.
  • To characterize this dynamics as a solution to a martingale problem.
  • To explore the convergence of finite graph models to continuous graphon dynamics.

Main Methods:

  • A Markov chain on finite graphs, inspired by the Moran model with resampling and mutation, was analyzed.
  • Finite graphs were encoded as graphemes (vertex set, adjacency matrix, sampling measure).
  • The space of graphons was equipped with convergence of sample subgraph densities.

Main Results:

  • The grapheme-valued Markov chain was shown to converge to a grapheme-valued diffusion as the number of vertices increases.
  • The resulting diffusion was characterized as a solution to a martingale problem.
  • A stationary distribution for the grapheme-valued diffusion was identified, linked to the Griffiths-Engen-McCloskey (GEM) distribution.

Conclusions:

  • The study provides a concrete example of population genetics dynamics on graphons as a diffusion process.
  • The findings establish a connection between discrete graph models and continuous graphon dynamics.
  • The stationary distribution offers insights into the long-term behavior of such genetic models.