[论文解读] Limits of multiplicative inhomogeneous random graphs and Lévy trees: Limit theorems

Nicolas Broutin, Thomas Duquesne+1 more|arXiv (Cornell University)|Feb 5, 2020

Stochastic processes and statistical mechanics参考文献 28被引用 13

一句话总结

本文通过 Lévy 过程和 Galton-Watson 森林嵌入，建立了乘法异质随机图向随机紧致测度度量空间的 Gromov-Hausdorff-Prokhorov 收敛性。该研究为临界参数区域提供了统一的极限理论，明确地从 Lévy 型过程的上升首达轨迹中构造极限对象，扩展了对 Erdős-Rényi、同质及幂律随机图的既有结果。

ABSTRACT

We consider a natural model of inhomogeneous random graphs that extends the classical Erd\H os-Rényi graphs and shares a close connection with the multiplicative coalescence, as pointed out by Aldous [AOP 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 [EJP 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. 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 regime-specific proofs, for instance: the case of Erd\H os-Rényi random graphs obtained by Addario-Berry, Goldschmidt and B. [PTRF 2012], the asymptotic homogeneous case as studied by Bhamidi, Sen and Wang [PTRF 2017], or the power-law case as considered by Bhamidi, Sen and van der Hofstad [PTRF 2018].

研究动机与目标

• 建立乘法异质随机图中连通分量的渐近度量结构。
• 统一临界参数区域下的极限定理，涵盖 Erdős-Rényi、同质及幂律模型。
• 提供一种使用 Lévy 过程和 Galton-Watson 森林嵌入的通用收敛框架。
• 从 Lévy 型过程的上升首达轨迹中显式构造极限对象。
• 扩展并推广文献中针对特定参数区域的既有证明。

提出的方法

• 通过捕捉乘法共alescence 动力学的 Lévy 过程对图进行编码。
• 将连通分量嵌入 Galton-Watson 森林中，渐近地转移至 Lévy 树的极限森林。
• 在 Gromov-Hausdorff-Prokhorov 拓扑下分析收敛性，目标为紧致测度度量空间。
• 极限对象从底层 Lévy 过程的上升首达轨迹中构造。
• 该方法依赖于 Aldous 和 Limic (1998) 提出的一般收敛条件，并在各类参数区域中统一应用。
• 该框架将极限结构统一嵌入到一个共同的概率模型中，从而统一并扩展了既有结果。

实验结果

研究问题

• RQ1 乘法异质随机图中连通分量的度量结构在渐近下如何表现？
• RQ2 是否能为异质随机图的临界参数区域推导出统一的极限定理？
• RQ3 如何从 Lévy 过程的上升首达轨迹中显式构造极限度量测度空间？
• RQ4 将图嵌入 Galton-Watson 森林后，在 Gromov-Hausdorff-Prokhorov 拓扑下如何传递至极限对象？
• RQ5 本研究结果在多大程度上推广了关于 Erdős-Rényi、同质及幂律随机图的既有发现？

主要发现

• 在 Gromov-Hausdorff-Prokhorov 拓扑下，乘法异质随机图的连通分量在分布上收敛于紧致测度度量空间。
• 极限对象从 Lévy 型过程的上升首达轨迹中显式构造，为极限提供了路径描述。
• 该收敛性在 Aldous 和 Limic (1998) 识别的一般临界参数区域条件下成立，统一了多个已知情形。
• 该方法在极限中将图的连通分量结构转移为 Lévy 树的森林，同时保持了度量与测度性质。
• 该框架扩展并推广了关于 Erdős-Rényi、同质及幂律随机图的既有结果，将它们统一于单一极限理论之下。
• 该方法提供了一种系统化的、非依赖于特定参数区域的证明策略，取代了早期的逐案分析方法。