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

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.

We focus on graphemes embedded in ultrametric spaces, where the ultrametric encodes the genealogy of the graph evolution, thereby drawing a direct connection to population genetics. The grapheme framework emphasises the embedding, in particular, in Polish spaces, and uses stronger notions of equivalence (homeomorphism and isometry) than the exchangeability underlying the Aldous–Hoover representation. We construct grapheme-valued Markov processes that arise as limits of finite graph evolutions, driven by rules analogous to the Fleming–Viot, Dawson–Watanabe and McKean–Vlasov processes from population genetics. We establish that these grapheme dynamics are characterised by well-posed martingale problems, leading to strong Markov processes with the Feller property and continuous paths (i.e., diffusions). We further derive duality relations by using coalescent processes, and identify the equilibria of dynamic graphemes, showing that these are linked to classical distributions arising in population genetics and can therefore be non-trivial.

Our approach extends and modifies previous work on graphon dynamics (Athreya et al., 2021), by providing a more general framework that includes a natural representation of the history of the graph. This allows for a rigorous treatment of the dynamics via martingale problems, and yields a characterisation of non-trivial equilibria.