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August 2021 Graphon-valued stochastic processes from population genetics
Siva Athreya, Frank den Hollander, Adrian Röllin
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Ann. Appl. Probab. 31(4): 1724-1745 (August 2021). DOI: 10.1214/20-AAP1631

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.

Funding Statement

SA was supported through an ISF-UGC grant and a CPDA from the Indian Statistical Institute. FdH was supported through NWO Gravitation Grant NETWORKS-024.002.003. AR was supported through Singapore Ministry of Education Academic Research Fund Tier 2 grant MOE2018-T2-2-076.

Acknowledgments

We thank two anonymous referees for their helpful comments. Part of this work was done while SA was visiting the Indian Institute of Science, Bangalore, and SA and FdH were visiting the International Center for Theoretical Sciences, Bangalore. The hospitality extended by these institutes is greatly appreciated.

Citation

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Siva Athreya. Frank den Hollander. Adrian Röllin. "Graphon-valued stochastic processes from population genetics." Ann. Appl. Probab. 31 (4) 1724 - 1745, August 2021. https://doi.org/10.1214/20-AAP1631

Information

Received: 1 September 2019; Revised: 1 May 2020; Published: August 2021
First available in Project Euclid: 15 September 2021

Digital Object Identifier: 10.1214/20-AAP1631

Subjects:
Primary: 05C80 , 60J68
Secondary: 60K35

Keywords: Fisher–Wright diffusion , Fleming–Viot diffusion , graphon dynamics , graphons , Moran model , Skorohod topology , subgraph densities

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 4 • August 2021
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