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

Abstract

This paper introduces graphemes for constructing and analyzing stochastic processes that describe the evolution of large dynamic graphs. Unlike graphons, which capture the static properties of dense graphs via exchangeability or subgraph densities, graphemes are capable of modeling the full space-time evolution of graphs. A grapheme is an equivalence class of triples: (Polish space, symmetric {0,1}-valued connection function, sampling probability measure). We focus on embeddings in ultrametric spaces, encoding the graph history and linking directly to population dynamics models. Graphemes utilize stronger equivalences (homeomorphism, isometry) than graphons. We construct grapheme-valued Markov processes as limits of finite graph evolutions, driven by Fleming-Viot, Dawson-Watanabe, and McKean-Vlasov analogues. We establish characterization via well-posed martingale problems, yielding strong Markov processes with the Feller property and continuous paths (diffusions). Duality relations involving coalescent processes are derived. We identify non-trivial equilibria, linked to classical distributions from population genetics. This framework extends [arXiv:1908.06241] by incorporating history, enabling rigorous analysis via martingale problems, and characterizing non-trivial long-term behavior.