Wasserstein distances between ERGMs and Erdős-Rényi models

TL;DR

This work investigates the microscopic distance between ferromagnetic ERGMs and Erdős--Rényi graphs by introducing the Hamming-Wasserstein distance and proving a sharp $\Theta(n^{3/2})$ scaling under an optimal coupling for parameter choices yielding metastable wells. The authors develop a triangle/wedge approximation to the discrete derivative of the Hamiltonian, enabling precise fluctuation bounds and a matching lower bound in all nondegenerate regimes, including ones with triangles. They combine metastable mixing, concentration, and an approximate FKG inequality to achieve tight upper bounds, and they provide an alternative lower-bound argument via a vertex-degree central limit theorem. The results quantify microscopic differences between ERGMs and ER graphs, offering robust techniques applicable to nonlinear graph models and Glauber-dynamics analyses in high-dimensional spin systems.

Abstract

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.

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1: Lemma 18 of bhamidi2008mixing
• Proposition 2.2: Proposition 5.9 of winstein2025concentration
• Lemma 2.3: Lemma 5.12 of winstein2025concentration
• Theorem 2.4: Theorem 1.1 of winstein2025concentration
• Theorem 2.5: Theorem 2.1 of barbour2022long, Corollary 4.4 of winstein2025concentration
• Lemma 2.6: Lemma 4.3 of winstein2025quantitative
• Proposition 2.7: Proposition 1.7 of winstein2025quantitative
• Corollary 2.8