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Continuum graph dynamics via population dynamics: Well-posedness, duality and equilibria

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  • Greven, Andreas
  • den Hollander, Frank
  • Klimovsky, Anton
  • Winter, Anita

Abstract

This paper introduces graphemes, a novel framework for constructing and analysing stochastic processes that describe the evolution of large dynamic graphs. Unlike graphons, which are well-suited for studying static dense graphs and which are closely related to the Aldous–Hoover representation of exchangeable random graphs, graphemes allow for a modelling of the full space–time evolution of dynamic dense graphs, beyond the exchangeability and the subgraph frequencies used in graphon theory. A grapheme is defined as an equivalence class of triples, consisting of a Polish space, a symmetric {0,1}-valued connection function on that space (representing edges connecting vertices), and a sampling probability measure.

Suggested Citation

  • Greven, Andreas & den Hollander, Frank & Klimovsky, Anton & Winter, Anita, 2025. "Continuum graph dynamics via population dynamics: Well-posedness, duality and equilibria," Stochastic Processes and their Applications, Elsevier, vol. 188(C).
    • Handle: RePEc:eee:spapps:v:188:y:2025:i:c:s0304414925001115
    DOI: 10.1016/j.spa.2025.104670
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    References listed on IDEAS

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    1. Aldous, David J., 1981. "Representations for partially exchangeable arrays of random variables," Journal of Multivariate Analysis, Elsevier, vol. 11(4), pages 581-598, December.
    2. Greven, Andreas & den Hollander, Frank & Klimovsky, Anton & Winter, Anita, 2024. "The grapheme-valued Wright–Fisher diffusion with mutation," Theoretical Population Biology, Elsevier, vol. 158(C), pages 76-88.
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