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

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

Stochastic Processes and their Applications, 2025, vol. 188, issue C

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.

Keywords: Graph-valued Markov processes; Graphemes; Marked graphemes; Duality; Martingale problem; Genealogy; Population genetics (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1016/j.spa.2025.104670

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