Ferromagnetic exponential random graph models (ERGMs) are random graph models under which the presence of certain small structures (such as triangles) is encouraged; they can be constructed by tilting an Erdős--Rényi model by the exponential of a particular nonlinear Hamiltonian. These models are mixtures of metastable wells which each behave macroscopically like an Erdős--Rényi model, exhibiting the same laws of large numbers for subgraph counts [CD13]. However, on the microscopic scale these metastable wells are very different from Erdős--Rényi models, with the total variation distance between the two measures tending to 1 [MX23]. In this article we clarify this situation by providing a sharp (up to constants) bound on the Hamming-Wasserstein distance between the two models, which is the average number of edges at which they differ, under the coupling which minimizes this average. In particular, we show that this distance is $Θ(n^{3/2})$, quantifying exactly how these models differ. An upper bound of this form has appeared in the past [RR19], but this was restricted to the subcritical (high-temperature) regime of parameters. We extend this bound, using a new proof technique, to the supercritical (low-temperature) regime, and prove a matching lower bound which has only previously appeared in the subcritical regime of special cases of ERGMs satisfying a "triangle-free" condition [DF25]. To prove the lower bound in the presence of triangles, we introduce an approximation of the discrete derivative of the Hamiltonian, which controls the dynamical properties of the ERGM, in terms of local counts of triangles and wedges (two-stars) near an edge. This approximation is the main technical and conceptual contribution of the article, and we expect it will be useful in a variety of other contexts as well. Along the way, we also prove a bound on the marginal edge probability under the ERGM via a new bootstrapping argument. Such a bound has already appeared [FLSW25], but again only in the subcritical regime and using a different proof strategy.

翻译：铁磁指数随机图模型（ERGMs）是一种随机图模型，它鼓励某些小型结构（如三角形）的出现；这类模型可以通过对Erdős-Rényi模型施加特定非线性哈密顿量的指数倾斜来构建。这些模型是多个亚稳态势阱的混合，每个势阱在宏观上表现得类似于Erdős-Rényi模型，呈现出相同的子图计数大数定律[CD13]。然而，在微观尺度上，这些亚稳态势阱与Erdős-Rényi模型差异显著，两者之间的总变差距离趋于1[MX23]。本文通过给出两个模型之间Hamming-Wasserstein距离的尖锐（常数因子内）界，澄清了这一情况。该距离是在最小化此平均值的耦合下，两个模型相异边的平均数量。具体而言，我们证明该距离为$Θ(n^{3/2})$，从而精确量化了这些模型的差异。此类上界在过去已有出现[RR19]，但仅限于参数的亚临界（高温）区域。我们采用新的证明技术，将此界扩展至超临界（低温）区域，并证明了一个匹配的下界，该下界先前仅出现在满足“无三角形”条件的特殊ERGM的亚临界区域[DF25]。为了在存在三角形的情况下证明下界，我们引入了哈密顿量离散导数的近似，该近似通过边附近的三角形和楔形（二星）的局部计数来控制ERGM的动态特性。这一近似是本文的主要技术和概念贡献，我们预期它也能在其他多种情境中发挥作用。在此过程中，我们还通过一种新的自举论证，证明了ERGM下边际边概率的一个界。此类界已经出现过[FLSW25]，但同样仅限于亚临界区域，且使用了不同的证明策略。