标题： 指数随机图的定量中心极限定理 显示英文标题

标题： Quantitative central limit theorems for exponential random graphs

摘要： 铁磁性指数随机图模型（ERGMs）是Erdős-Rényi模型的非线性指数倾斜，在这种模型下，如三角形等某些子图的存在可能被强调。 这些模型是亚稳井的混合体，每个亚稳井在宏观上都像新的Erdős-Rényi模型，表现出相同的总体边数以及所有子图计数的大数定律。 然而，这些量的微观波动在一段时间内仍然难以捉摸。 在Fang、Liu、Shao和Zhao [FLSZ24] 的最新突破基础上，该突破由Stein方法推动，我们证明了在铁磁性ERGM的亚稳井中，这些量及其更多内容的定量中心极限定理（CLTs）。 我们结果的主要新意在于它们也适用于参数的超临界（低温）区域，这一区域此前相对未被探索。 为了实现这一点，我们开发了一种基于ERGM Glauber动力学下相关量演化的细致分析的新概率技术。 我们的技术使我们能够提供[FLSZ24]开发的方法的主要输入，即子图计数的波动是由总体边数的波动驱动的。 Sambale和Sinulis [SS20] 首次在Dobrushin（极高温度）区域通过函数分析方法证明了这一点。 我们认为我们的技术阐明了其中起作用的基本机制，并且与[FLSZ24]在Dobrushin区域之外的亚临界（高温）区域的结果相比，还提供了观测量与极限高斯分布之间Wasserstein和Kolmogorov距离的改进界限。 此外，我们的技术足够灵活，还可以得出顶点度数和局部子图计数的定量CLTs，这些在任何参数区域中以前都没有出现过。 显示英文摘要

摘要： Ferromagnetic exponential random graph models (ERGMs) are nonlinear exponential tilts of Erd\H{o}s-R\'enyi models, under which the presence of certain subgraphs such as triangles may be emphasized. These models are mixtures of metastable wells which each behave macroscopically like new Erd\H{o}s-R\'enyi models themselves, exhibiting the same laws of large numbers for the overall edge count as well as all subgraph counts. However, the microscopic fluctuations of these quantities remained elusive for some time. Building on a recent breakthrough by Fang, Liu, Shao and Zhao [FLSZ24] driven by Stein's method, we prove quantitative central limit theorems (CLTs) for these quantities and more in metastable wells under ferromagnetic ERGMs. One main novelty of our results is that they apply also in the supercritical (low temperature) regime of parameters, which has previously been relatively unexplored. To accomplish this, we develop a novel probabilistic technique based on the careful analysis of the evolution of relevant quantities under the ERGM Glauber dynamics. Our technique allows us to deliver the main input to the method developed by [FLSZ24], which is the fact that the fluctuations of subgraph counts are driven by those of the overall edge count. This was first shown for the triangle count by Sambale and Sinulis [SS20] in the Dobrushin (very high temperature) regime via functional-analytic methods. We feel our technique clarifies the underlying mechanisms at play, and it also supplies improved bounds on the Wasserstein and Kolmogorov distances between the observables at hand and the limiting Gaussians, as compared to the results of [FLSZ24] in the subcritical (high temperature) regime beyond the Dobrushin regime. Moreover, our technique is flexible enough to also yield quantitative CLTs for vertex degrees and local subgraph counts, which have not appeared before in any parameter regime.