标题： 通过种群动力学研究连续图的动力学：适定性、对偶性和平衡态 显示英文标题

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

摘要： 本文介绍了图元的概念，用于构建和分析描述大规模动态图演化的随机过程。与图限（graphons）不同，后者通过可交换性或子图密度捕捉稠密图的静态属性，图元能够建模图的全时空演化。一个图元是一个三元组等价类：(波莱尔空间，对称的{0,1}-值连接函数，抽样概率测度)。我们专注于嵌入超度量空间，编码图的历史，并直接与种群动力学模型相联系。图元利用比图限更强的等价关系（同胚、等距）。我们构造了以有限图演化极限为驱动的图元值马尔可夫过程，这些过程由类似于Fleming-Viot、Dawson-Watanabe和McKean-Vlasov的过程驱动。我们通过适定的鞅问题表征得到了具有福勒性质和连续路径（扩散）的强马尔可夫过程。推导出了涉及凝聚过程的对偶关系。我们识别了非平凡平衡态，这些平衡态与种群遗传学中的经典分布相关联。此框架扩展了[arXiv:1908.06241]，通过纳入历史，允许通过鞅问题进行严格的分析，并刻画非平凡的长期行为。 显示英文摘要

摘要： 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.