[论文解读] Limits of multiplicative inhomogeneous random graphs and Levy trees

Nicolas Broutin, Thomas Duquesne+1 more|arXiv (Cornell University)|Apr 21, 2018

Stochastic processes and statistical mechanics参考文献 11被引用 13

一句话总结

本文研究具有顶点权重的非齐次随机图，该权重控制边的形成，通过 Lévy 过程编码和嵌入到 Galton-Watson 森林中，证明了连通分量度量结构在 Gromov-Hausdorff-Prokhorov 意义下收敛于随机紧致测度度量空间。关键贡献在于，从 Lévy 型过程的上升首出（excursions）出发，统一且显式地构造了极限对象，并给出了清晰的紧致性条件以及分形维数的确定方法。

ABSTRACT

We consider a natural model of inhomogeneous random graphs that extends the classical Erdos-Renyi graphs and shares a close connection with the multiplicative coalescence, as pointed out by Aldous . 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 have identified the entrance boundary of the multiplicative coalescence, which is intimately related to the excursion lengths of certain Levy-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. 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 Levy 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 Levy trees, which allows us to give an explicit construction of the limit objects from the excursions of the Levy-type process. As a consequence of our construction, we give a transparent and explicit condition for the compactness of the limit objects and determine their fractal dimensions. These results extend and complement several previous results that had obtained via model- or regime-specific proofs, for instance: the case of Erdos-Renyi random graphs obtained by Addario-Berry, Goldschmidt and B., the asymptotic homogeneous case as studied by Bhamidi, Sen and Wang, or the power-law case as considered by Bhamidi, Sen and van der Hofstad.

研究动机与目标

• 理解具有顶点权重的非齐次随机图的渐近度量结构。
• 在单一框架下统一现有不同参数区域（如 Erdos-Renyi、幂律、齐次）下关于连通分量收敛结果。
• 从 Lévy 型过程的上升首出显式构造极限度量测度空间。
• 推导极限对象紧致性的必要且充分条件。
• 确定极限随机紧致度量空间的分形维数。

提出的方法

• 通过 Lévy 过程编码随机图，以捕捉连通分量的质量与度量结构。
• 将图的连通分量嵌入到 Galton-Watson 森林中，以传递渐近行为。
• 证明嵌入后的图分量与 Lévy 树森林的分量在渐近意义上等价。
• 利用 Lévy 过程的首出理论，从首出显式构造极限对象。
• 应用 Gromov-Hausdorff-Prokhorov 收敛性，在紧致测度度量空间中建立极限。
• 通过 Lévy 过程的标度性质推导紧致性条件并计算分形维数。

实验结果

研究问题

• RQ1 在一般临界参数区域下，非齐次随机图中连通分量的度量结构如何收敛？
• RQ2 Lévy 型过程的上升首出与图分量的极限度量测度空间之间存在何种显式关系？
• RQ3 在何种条件下极限对象是紧致的？
• RQ4 如何从底层的 Lévy 过程确定极限空间的分形维数？
• RQ5 能否建立一个统一框架，通过单一构造方法恢复以往针对特定参数区域（如 Erdos-Renyi、幂律）的结果？

主要发现

• 在 Gromov-Hausdorff-Prokhorov 拓扑下，非齐次随机图的连通分量以分布收敛于一个随机紧致测度度量空间。
• 极限对象从 Lévy 型过程的上升首出显式构造，提供了清晰且统一的表示形式。
• 基于 Lévy 过程的性质，推导出极限对象紧致性的必要且充分条件。
• 极限空间的分形维数由 Lévy 过程的标度指数决定，推广了特定参数区域下的已知结果。
• 该方法通过单一连贯的框架，恢复并扩展了关于 Erdos-Renyi、齐次及幂律非齐次随机图的先前结果。
• 嵌入到 Galton-Watson 森林中的渐近行为转移至嵌入到 Lévy 树森林中，从而实现极限对象的显式构造。