Concentration via metastable mixing, with applications to the supercritical exponential random graph model

TL;DR

This work rigorously extends concentration of measure for exponential random graphs into the metastable, supercritical regime by leveraging metastable mixing within wells around constant graphons $W_{p^*}$. It introduces a novel Lipschitz-concentration inequality tailored to metastable dynamics, using inputs from recent metastable mixing results and Barbour’s framework to handle global observables, plus an exchangeable-pairs approach for local observables. The results yield quantitative bounds on the Wasserstein distance to Erdős–Rényi graphs, a central limit theorem for small edge subcollections, and normal approximations for edge counts in subgraphs, all under metastable conditioning. Simulations illustrate metastable wells and validate the predicted scaling laws, highlighting the practical impact for understanding microscopic fluctuations in low-temperature ERGMs and guiding sampling in phase-coexisting regimes.

Abstract

Folklore belief holds that metastable wells in low-temperature statistical mechanics models exhibit high-temperature behavior. We make this rigorous in the exponential random graph model (ERGM) through the lens of concentration of measure. We make use of the supercritical (low-temperature) metastable mixing which was recently proven by Bresler, Nagaraj, and Nichani, and obtain a novel concentration inequality for Lipschitz observables of the ERGM in a large metastable well, answering a question posed by those authors. To achieve this, we prove a new connectivity property for metastable mixing in the ERGM and introduce a new general result yielding concentration inequalities, which extends a result of Chatterjee. We also use a result of Barbour, Brightwell, and Luczak to cover all cases of interest. Our work extends a result of Ganguly and Nam from the subcritical (high-temperature) regime to metastable wells, and we also extend applications of this concentration, namely a central limit theorem for small subcollections of edges and a bound on the Wasserstein distance between the ERGM and the Erdős-Rényi random graph. Finally, to supplement the mathematical content of the article, we present a simulation study of metastable wells in the supercritical ERGM.

Figures (7)

• Figure 1: Three views of a numerical plot of the maximizer $p^*$ of $L_{\bm\beta}$, as a function of $\beta_0 \in [-2,1]$ and $\beta_1, \beta_2 \in [0,2]$. The graphs chosen for this ERGM are $G_1 =$ two-star (or disjoint union of two edges), and $G_2 =$ triangle (or any other graph on $3$ edges, as all give the same values of $p^*$). The opacity of $p^*$ values near $0$ and $1$ has been decreased so that the interior detail can be seen: notice the two-dimensional surface across which there is a first-order phase transition.
• Figure 2: Three views of the two parameter regimes in the ERGM with the same specification as in Figure \ref{['fig:ergm_pstar']}. Top row: subcritical regime. Bottom row: supercritical regime.
• Figure 3: Plots of $L_{\bm\beta}$ for various ERGM specifications, as well as marks representing the datasets we consider in Section \ref{['sec:simulations']}. All specifications have $K=2$ and $G_0, G_1$ as an edge and a $2$-star (wedge). For the lower blue curve, $G_2$ is a triangle, and for the upper red curve, $G_2$ is either a tetrahedron or a hexagon, both of which yield the same $L_{\bm\beta}$. The local maxima have been highlighted and labeled with symbols corresponding to the datasets outlined in Table \ref{['table:data']}. For the lower blue curve, the global maximum is on the left, while for the upper red curve, the global maximum is on the right. We also remark that it is possible for $L_{\bm\beta}$ to have more than two local maxima; the green curve without any marked points demonstrates this, corresponding to the specification where $G_2$ is any graph with $32$ edges and we take $\bm\beta = (-1.17, 1.17, 0.06)$. However, due to computational constraints, we did not generate a dataset for this specification.
• Figure 4: Log-log plots of the standard deviations of four Lipschitz observables of ERGM samples. The power-law dependences agree with the guarantees of Theorem \ref{['thm:lipschitzconcentration_informal']}. Note that the somewhat degenerate behavior of datasets 4 and 6 likely arises from the fact that $p^*$ is very close to $1$.
• Figure 5: Left: a $64$-vertex ERGM sample from dataset 1, overlaid with a coupled sample from $\mathcal{G}(64,p^*)$. Grey edges appear in both graphs, blue edges appear only in the ERGM, and orange edges appear only in the Erdős--Rényi graph. Right: the graph structure of the discrepancy edges. Notice that at first glance it appears more clustered than one might expect from a typical graph with this many vertices and edges.

Theorems & Definitions (41)

• Theorem 1.1: Informal, see Theorem \ref{['thm:lipschitzconcentration']}
• Theorem 1.2: Informal, see Theorem \ref{['thm:wasserstein']}
• Theorem 1.3: Informal, see Theorem \ref{['thm:clt']}
• Conjecture 2.1: Resolved!
• Conjecture 2.2
• Theorem 3.1: Theorems 3.2 and 4.1 of chatterjee2013estimating
• Theorem 3.2: Theorem 1 of ganguly2024sub
• Theorem 4.1: Theorem 3.3 of chatterjee2005concentration
• Theorem 4.2
• proof : Proof of Theorem \ref{['thm:chatterjee']}