We consider a natural model of inhomogeneous random graphs that extends the classical Erdős–Rényi graphs and shares a close connection with the multiplicative coalescence, as pointed out by Aldous (Ann Probab 25:812–854, 1997). In this model, the vertices are assigned weights that govern their tendency to form edges. It is by looking at the asymptotic distributions of the masses (sum of the weights) of the connected components of these graphs that Aldous and Limic (Electron J Probab 3:1–59, 1998) have identified the entrance boundary of the multiplicative coalescence, which is intimately related to the excursion lengths of certain Lévy-type processes. We, instead, look at the metric structure of these components and prove their Gromov–Hausdorff–Prokhorov convergence to a class of (random) compact measured metric spaces that have been introduced in a companion paper (Broutin et al. in Limits of multiplicative inhomogeneous random graphs and Lévy trees: the continuum graphs. arXiv:1804.05871, 2020). Our asymptotic regimes relate directly to the general convergence condition appearing in the work of Aldous and Limic. Our techniques provide a unified approach for this general “critical” regime, and relies upon two key ingredients: an encoding of the graph by some Lévy process as well as an embedding of its connected components into Galton–Watson forests. This embedding transfers asymptotically into an embedding of the limit objects into a forest of Lévy trees, which allows us to give an explicit construction of the limit objects from the excursions of the Lévy-type process. The mains results combined with the ones in the other paper allow us to extend and complement several previous results that had been obtained via model- or regime-specific proofs, for instance: the case of Erdős–Rényi random graphs obtained by Addario-Berry et al. (Probab Theory Relat Fields 152:367–406, 2012), the asymptotic homogeneous case as studied by Bhamidi et al. (Probab Theory Relat Fields 169:565–641, 2017), or the power-law case as considered by Bhamidi et al. (Probab Theory Relat Fields 170:387–474, 2018).

正如 Aldous (Ann Probab 25:812-854, 1997) 所指出的，我们考虑了非齐次随机图的自然模型，该模型扩展了经典的 Erdős-Rényi 图并与乘法合并有着密切的联系。在这个模型中，顶点被分配权重来控制它们形成边的趋势。Aldous 和 Limic (Electron J Probab 3:1–59, 1998) 正是通过查看这些图的连接分量的质量（权重之和）的渐近分布，确定了乘法合并的入口边界，这与某些 Lévy 型过程的偏移长度密切相关。相反，我们 查看这些组件的度量结构，并证明它们的 Gromov-Hausdorff-Prokhorov 收敛到一类（随机）紧致度量空间，这些空间已在一篇配套论文中引入（Broutin 等人在 Limits of multiplicative inhomogeneous random graphs 和 Lévy树：连续图。arXiv：1804.05871，2020）。我们的渐近机制与出现在 Aldous 和 Limic 工作中的一般收敛条件直接相关。我们的技术为这种一般的“关键”机制提供了统一的方法，并依赖于两个关键要素：通过一些 Lévy 过程对图进行编码，以及将其连接组件嵌入到高尔顿-沃森森林中。这种嵌入逐渐转移到限制对象嵌入 Lévy 树的森林中，这使我们能够从 Lévy 型过程的偏移中给出极限对象的显式构造。主要结果与另一篇论文中的结果相结合，使我们能够扩展和补充之前通过特定模型或特定制度证明获得的几个结果，例如：Addario-Berry 等人获得的 Erdős-Rényi 随机图的情况阿尔。(Probab Theory Relat Fields 152:367–406, 2012)，Bhamidi 等人研究的渐近齐次案例。（Probab Theory Relat Fields 169:565–641, 2017），或Bhamidi 等人考虑的幂律案例。(Probab Theory Relat Fields 170:387–474, 2018)。