Ferromagnetic ERGMs: Theory and Phase Behavior

• Ferromagnetic ERGMs are Gibbs measures on graphs that reward the appearance of specific subgraphs through nonnegative interaction parameters.
• They are analyzed using variational free energies and phase regime techniques to uncover metastability, phase transitions, and fluctuation dynamics.
• Extensions such as sparse fractional-power and block-structured models enhance understanding by comparing these measures with classical Erdős–Rényi benchmarks.

Searching arXiv for the cited ferromagnetic ERGM papers to ground the article in current literature. Ferromagnetic exponential random graph models (ERGMs) are Gibbs measures on labeled graphs in which the presence of specified subgraphs is rewarded by nonnegative interaction parameters. In the standard homogeneous formulation, one fixes finite simple graphs $G_0,G_1,\dots,G_K$, with $G_0$ the single edge, defines the homomorphism density $t(G_j,x)=N_{G_j}(x)/n^{v_j}$, chooses $\beta_0\in\mathbb R$ and $\beta_1,\dots,\beta_K\ge 0$, and sets

$$H(x)=\sum_{j=0}^K \beta_j\,t(G_j,x),\qquad \mathbb P[X=x]\propto \exp(n^2 H(x)).$$

Equivalently, this is an exponential tilt of $G(n,q_0)$ with $q_0=e^{2\beta_0}/(1+e^{2\beta_0})$ (Winstein, 14 Jul 2025). A sparse fractional-power variant fixes simple graphs $F_1,\dots,F_m$ and exponents $\alpha_k\in(0,1)$, defines

$G_0$0

and assigns Gibbs weight

$G_0$1

with $G_0$2 (Cook et al., 2022). Across these formulations, ferromagnetic ERGMs are analyzed through variational free energies, phase structure, large deviations, metastable dynamics, and graphon methods; recent work also gives sharp fluctuation and transport results, as well as block-structured extensions (Winstein, 20 Jan 2026, Magnanini, 18 Feb 2026).

1. Definitions and model classes

In the homogeneous ferromagnetic model, the state space is the set of all simple graphs on $G_0$3, and the nonlinear terms $G_0$4 reward the appearance of copies of $G_0$5 when $G_0$6. The single-edge term $G_0$7 plays the role of an external field, while higher-order terms such as triangles and two-stars enter as positive interactions. The measure may also be written

$G_0$8

or, equivalently,

$G_0$9

where $t(G_j,x)=N_{G_j}(x)/n^{v_j}$0 is the raw homomorphism count (Winstein, 20 Jan 2026).

The sparse fractional-power model is formulated on

$t(G_j,x)=N_{G_j}(x)/n^{v_j}$1

with homomorphism density

$t(G_j,x)=N_{G_j}(x)/n^{v_j}$2

Here $t(G_j,x)=N_{G_j}(x)/n^{v_j}$3 gives “vanishing edge density,” so that typical $t(G_j,x)=N_{G_j}(x)/n^{v_j}$4. The key modification is the fractional power $t(G_j,x)=N_{G_j}(x)/n^{v_j}$5: by taking $t(G_j,x)=N_{G_j}(x)/n^{v_j}$6, “the penalization for large subgraph counts grows sublinearly, preventing the abrupt jump to the complete graph seen in the standard $t(G_j,x)=N_{G_j}(x)/n^{v_j}$7 edge–triangle model” (Cook et al., 2022).

A further inhomogeneous class is the block-structured edge–triangle ERGM. With a partition $t(G_j,x)=N_{G_j}(x)/n^{v_j}$8, block sizes $t(G_j,x)=N_{G_j}(x)/n^{v_j}$9, triangle parameters $\beta_0\in\mathbb R$0, and edge parameters $\beta_0\in\mathbb R$1, the Hamiltonian is

$\beta_0\in\mathbb R$2

with $\beta_0\in\mathbb R$3 for $\beta_0\in\mathbb R$4 (Magnanini, 18 Feb 2026).

2. Variational structure and phase regimes

For homogeneous ferromagnetic ERGMs, the basic mean-field functional is

$\beta_0\in\mathbb R$5

Its global maximizers $\beta_0\in\mathbb R$6 govern macroscopic laws of large numbers: any ERGM sample $\beta_0\in\mathbb R$7 satisfies $\beta_0\in\mathbb R$8 for each $\beta_0\in\mathbb R$9, for some $\beta_1,\dots,\beta_K\ge 0$0. Glauber-dynamics analysis introduces

$\beta_1,\dots,\beta_K\ge 0$1

and any attracting fixed point $\beta_1,\dots,\beta_K\ge 0$2 of $\beta_1,\dots,\beta_K\ge 0$3 is a strictly concave local maximum of $\beta_1,\dots,\beta_K\ge 0$4. The regimes stated in the recent fluctuation theory are: Dobrushin (very-high-temperature), where $\beta_1,\dots,\beta_K\ge 0$5; subcritical (high-temperature), where $\beta_1,\dots,\beta_K\ge 0$6 has a unique attracting fixed point; and supercritical (low-temperature), where $\beta_1,\dots,\beta_K\ge 0$7 has multiple attracting fixed points, with “large-scale metastability and slow mixing from worst-case starts but rapid (metastable) mixing within each ‘well’.” Phase coexistence occurs where $\beta_1,\dots,\beta_K\ge 0$8, but one may condition on the well around a given $\beta_1,\dots,\beta_K\ge 0$9 to recover uniqueness (Winstein, 14 Jul 2025).

In the sparse fractional-power model, the partition function is

$$H(x)=\sum_{j=0}^K \beta_j\,t(G_j,x),\qquad \mathbb P[X=x]\propto \exp(n^2 H(x)).$$0

The Gibbs variational principle gives

$$H(x)=\sum_{j=0}^K \beta_j\,t(G_j,x),\qquad \mathbb P[X=x]\propto \exp(n^2 H(x)).$$1

Restricting $$H(x)=\sum_{j=0}^K \beta_j\,t(G_j,x),\qquad \mathbb P[X=x]\propto \exp(n^2 H(x)).$$2 to product laws $$H(x)=\sum_{j=0}^K \beta_j\,t(G_j,x),\qquad \mathbb P[X=x]\propto \exp(n^2 H(x)).$$3 and replacing $$H(x)=\sum_{j=0}^K \beta_j\,t(G_j,x),\qquad \mathbb P[X=x]\propto \exp(n^2 H(x)).$$4 yields the naïve mean-field free energy

$$H(x)=\sum_{j=0}^K \beta_j\,t(G_j,x),\qquad \mathbb P[X=x]\propto \exp(n^2 H(x)).$$5

where $$H(x)=\sum_{j=0}^K \beta_j\,t(G_j,x),\qquad \mathbb P[X=x]\propto \exp(n^2 H(x)).$$6 and

$$H(x)=\sum_{j=0}^K \beta_j\,t(G_j,x),\qquad \mathbb P[X=x]\propto \exp(n^2 H(x)).$$7

If one further restricts to homogeneous $$H(x)=\sum_{j=0}^K \beta_j\,t(G_j,x),\qquad \mathbb P[X=x]\propto \exp(n^2 H(x)).$$8, then

$$H(x)=\sum_{j=0}^K \beta_j\,t(G_j,x),\qquad \mathbb P[X=x]\propto \exp(n^2 H(x)).$$9

with

$G(n,q_0)$0

The derivation outline is: apply Varadhan’s lemma to the LDP for homomorphism densities in $G(n,q_0)$1, exchange supremum and limit, and observe that the extremum is reached on constant-$G(n,q_0)$2 (Cook et al., 2022).

3. Sparse ferromagnetic structure, upper tails, and nondegeneracy

The sparse theory identifies a two-parameter structural description of upper-tail events and of the corresponding ERGM. For thresholds $G(n,q_0)$3, define

$G(n,q_0)$4

and let $G(n,q_0)$5. Under $G(n,q_0)$6 and $G(n,q_0)$7, if $G(n,q_0)$8 solves the planar variational problem

$G(n,q_0)$9

with

$q_0=e^{2\beta_0}/(1+e^{2\beta_0})$0

then with high probability

$q_0=e^{2\beta_0}/(1+e^{2\beta_0})$1

and, in a refined regime,

$q_0=e^{2\beta_0}/(1+e^{2\beta_0})$2

Here $q_0=e^{2\beta_0}/(1+e^{2\beta_0})$3 imposes an almost-clique of size $q_0=e^{2\beta_0}/(1+e^{2\beta_0})$4 plus almost-biclique (hub) of size $q_0=e^{2\beta_0}/(1+e^{2\beta_0})$5, and $q_0=e^{2\beta_0}/(1+e^{2\beta_0})$6 imposes a spectral-norm neighborhood of the corresponding weighted “clique-hub” graph $q_0=e^{2\beta_0}/(1+e^{2\beta_0})$7 (Cook et al., 2022).

The same two-parameter family appears in the upper-tail LDP. The stated rate is

$q_0=e^{2\beta_0}/(1+e^{2\beta_0})$8

with

$q_0=e^{2\beta_0}/(1+e^{2\beta_0})$9

and the joint problem for $F_1,\dots,F_m$0 reduces to

$F_1,\dots,F_m$1

Hence “typical ERGMs have two phases: clique-driven ($F_1,\dots,F_m$2) or hub-driven ($F_1,\dots,F_m$3)” (Cook et al., 2022).

The analytic input singled out in this work is a stability form of Finner’s generalized Hölder inequality. Classical Finner gives

$F_1,\dots,F_m$4

under $F_1,\dots,F_m$5. The stability statement says that if additionally $F_1,\dots,F_m$6, then there exist nonnegative $F_1,\dots,F_m$7 with $F_1,\dots,F_m$8 such that

$F_1,\dots,F_m$9

for every $\alpha_k\in(0,1)$0, $\alpha_k\in(0,1)$1. Combined with quantitative LDPs, this identifies near-optimizers of the free-energy variational problem as clique-hub graphons (Cook et al., 2022).

4. Metastability and quantitative fluctuation theory

Ferromagnetic ERGMs 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.” The fluctuation theory developed in 2025 works in the noncritical “phase-uniqueness” case $\alpha_k\in(0,1)$2 or, in phase coexistence, after conditioning on a single well. If $\alpha_k\in(0,1)$3 is the unique density, then for the edge count

$\alpha_k\in(0,1)$4

and for every $\alpha_k\in(0,1)$5,

$\alpha_k\in(0,1)$6

For a fixed graph $\alpha_k\in(0,1)$7 with $\alpha_k\in(0,1)$8 vertices and $\alpha_k\in(0,1)$9 edges,

$G_0$00

and the same $G_0$01 rate holds after the corresponding normalization. For any fixed vertex $G_0$02, the degree satisfies a CLT with variance

$G_0$03

and both Wasserstein and Kolmogorov distances are $G_0$04. Local subgraph counts satisfy analogous CLTs with the same $G_0$05 rate (Winstein, 14 Jul 2025).

The main probabilistic mechanism is a combination of Stein’s method for nonlinear exponential families with a new “Hájék-projection” analysis via Glauber-dynamics-based concentration. The global proposition states that for

$G_0$06

one has, for any $G_0$07,

$G_0$08

and in particular $G_0$09. The local version gives, for $G_0$10,

$G_0$11

with $G_0$12 (Winstein, 14 Jul 2025).

These results extend quantitative CLTs to the “full supercritical (low-temperature) regime (including metastable cases),” improve the error to $G_0$13 for edges and global subgraphs, and provide the first quantitative CLTs for vertex degrees and local subgraph counts (Winstein, 14 Jul 2025).

5. Erdős–Rényi comparison and Hamming–Wasserstein geometry

A central structural point is that phase-conditioned ferromagnetic ERGMs are close to $G_0$14 at one scale and far from it at another. In the ferromagnetic regime, the free-energy functional

$G_0$15

has a finite set $G_0$16 of global maximizers, and the ERGM law is, up to total-variation error $G_0$17, a finite mixture of phase measures $G_0$18, $G_0$19. For each such $G_0$20, if $G_0$21 and $G_0$22, then their cut-distance goes to zero, but statistical tests can distinguish the laws with vanishing error probability, forcing

$G_0$23

The Hamming–Wasserstein distance resolves this discrepancy: $G_0$24 Under $G_0$25 and the nondegeneracy assumption that “at least one $G_0$26 is not a disjoint union of edges,” there are constants $G_0$27 such that

$G_0$28

so the distance is $G_0$29 (Winstein, 20 Jan 2026).

The upper bound is obtained by monotone Glauber-dynamics coupling and contraction in a “good set.” The lower bound uses the optimal-coupling characterization together with the conditional edge-update

$G_0$30

The key approximation expresses $G_0$31 through local triangle and wedge counts near $G_0$32: $G_0$33 where $G_0$34 and $G_0$35 are nonnegative combinations of the model parameters, and at least one is positive whenever the model is nondegenerate. This wedge–triangle linearization yields $G_0$36, and summing over all edges gives the $G_0$37 lower bound (Winstein, 20 Jan 2026).

The same analysis also gives a bound on the marginal edge probability,

$G_0$38

via a bootstrapping argument. A standard misunderstanding is therefore avoided: cut-distance and laws of large numbers describe the macroscopic resemblance to $G_0$39, whereas Hamming–Wasserstein and total variation detect genuine microscopic separation (Winstein, 20 Jan 2026).

6. Block-structured ferromagnetic ERGMs

The block-model extension places ERGMs on a vertex partition with type-dependent interaction parameters. The Gibbs measure

$G_0$40

is pushed forward to the space of $G_0$41-colored graphons. Writing

$G_0$42

$G_0$43

and

$G_0$44

the measures satisfy an LDP with rate function

$G_0$45

and the limiting free energy is

$G_0$46

(Magnanini, 18 Feb 2026).

In the ferromagnetic block regime, $G_0$47 for all $G_0$48. By a block-wise Hölder argument, any maximizer may be chosen to be a $G_0$49-block-constant graphon

$G_0$50

This reduces the graphon variational problem to a finite-dimensional optimization over the matrix entries $G_0$51. The fact that every optimizer is block-constant is identified as the natural analogue of the usual “replica-symmetric” regime in the homogeneous model (Magnanini, 18 Feb 2026).

A Dobrushin-type uniqueness criterion is obtained from the map

$G_0$52

for which

$G_0$53

Hence $G_0$54 implies that $G_0$55 is a contraction, so there is a unique fixed point $G_0$56, which is the unique maximizer of the finite-dimensional variational problem and therefore of the original infinite-dimensional problem. Under the same condition,

$G_0$57

Equivalently, under the unique Gibbs measure on the infinite vertex set, the empirical block-pair edge density converges to the entries of $G_0$58 (Magnanini, 18 Feb 2026).