Quantitative central limit theorems for exponential random graphs

TL;DR

This work establishes quantitative central limit theorems for edge counts, subgraph counts, vertex degrees, and local subgraph counts in ferromagnetic ERGMs, including the supercritical regime where metastable wells govern fluctuations. By combining Hájek projections with concentration via mixing and a Stein's method framework for nonlinear exponential families, the authors derive explicit Wasserstein and Kolmogorov distance bounds that improve prior subcritical results and extend to phase-coexistence settings. A novel modified Hamiltonian concentrates the analysis near metastable wells, enabling precise control of error terms and cancellations that yield Gaussian fluctuations with explicit rates. The results significantly advance the understanding of microscopic fluctuations in ERGMs and provide quantitative tools for inference and prediction across parameter regimes, including local observables and subgraph counts. Overall, the paper delivers a comprehensive, rigorously quantified picture of Gaussian fluctuations in dense ERGMs under ferromagnetic interaction across multiple regimes, with broad methodological implications for probabilistic graphical models.

Abstract

Ferromagnetic exponential random graph models (ERGMs) are nonlinear exponential tilts of Erdős-Rényi 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 Erdős-Rényi 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.

Figures (3)

• Figure 1: Plot of $L_\beta(q)$, as defined in \ref{['eq:Ldef']}, for the ERGM with $G_0, G_1, G_2$ an edge, a wedge ($2$-star), and a triangle respectively, with parameters $\beta = (\beta_0,\beta_1,\beta_2) = (-1, 0.53, 0.5)$. The unique optimal density $p$, which maximizes $L_\beta$ globally, is highlighted. There are multiple local maximizers, so this choice of $\beta$ lies in the supercritical or low-temperature regime for this ERGM, which will be introduced in Section \ref{['sec:review_fluctuations_dynamical']}.
• Figure 2: Plots of the three relevant parameter regimes, for the same ERGM as in Figure \ref{['fig:lbeta']} (with $G_0,G_1,G_2$ being an edge, wedge, and triangle) as $\beta = (\beta_0,\beta_1,\beta_2)$ varies in $[-2,1] \times [0,2] \times [0,2]$. The colors indicate the value of $p$, the optimizer of $L_\beta$, with purple denoting $p=0$ and red denoting $p=1$, other colors interpolating between these two extremes. Left: Dobrushin or "very high temperature" regime, where the natural dynamics exhibit uniform contraction. Middle: subcritical or "high temperature" regime, where the dynamics exhibit rapid mixing. Right: supercritical or "low temperature" regime, where the dynamics mix slowly from a worst-case initialization but exhibit metastable mixing from a warm start within a metastable well. The supercritical regime also includes the phase transition surface, on which there is phase coexistence.
• Figure 3: Plot of $\varphi_\beta(q)$, as defined in \ref{['eq:phidef1']}, for the ERGM with the same specifying graphs and parameters as in Figure \ref{['fig:lbeta']}. Compare with that figure, which depcits a plot of $L_\beta(q)$, and observe that the attracting fixed points of $\varphi_\beta$ are the same as the local maxima of $L_\beta$. In particular, the global maximum $p$ of $L_\beta$ is an attracting fixed point, but there may be other attracting fixed points of $\varphi_\beta$ which are not global maxima of $L_\beta$.

Theorems & Definitions (45)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Corollary 1.4
• Proposition 1.5
• Proposition 1.6
• Proposition 1.7
• Theorem 2.1: Theorem 3.2 of chatterjee2013estimating
• Theorem 2.2: Theorem 4.1 of chatterjee2013estimating
• Theorem 2.3: Theorem 5.1 of winstein2025concentration