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ERGMs on block models

Elena Magnanini

TL;DR

This work extends exponential random graph models to inhomogeneous, block-structured settings by introducing a finite partition of vertices into $k$ blocks and a block-aware edge-triangle Hamiltonian. It develops a colored graphon framework and proves a large deviation principle for the associated measures, yielding a variational formula for the limiting free energy. In the ferromagnetic regime where triangle weights are nonnegative, the variational problem collapses to a finite-dimensional scalar fixed-point system for the block-connection matrix, providing a block-wise replica-symmetric analogue. Under a Dobrushin-type condition, the maximizer is unique and a strong law of large numbers holds for the block-edge density, establishing precise asymptotic behavior and a clear route to studying symmetry-breaking phenomena in more general regimes.

Abstract

We extend the classical edge-triangle Exponential Random Graph Model (ERGM) to an inhomogeneous setting in which vertices carry types determined by an underlying partition. This leads to a block-structured ERGM where interaction parameters depend on vertex types. We establish a large deviation principle for the associated sequence of measures and derive the corresponding variational formula for the limiting free energy. In the ferromagnetic regime, where the parameters governing triangle densities are nonnegative, we reduce the variational problem to a scalar optimization problem, thereby identifying the natural block counterpart of the replica symmetric regime. Under additional restrictions on the parameters, comparable to the classical Dobrushin's uniqueness region, we prove uniqueness of the maximizer and derive a law of large numbers for the edge density.

ERGMs on block models

TL;DR

This work extends exponential random graph models to inhomogeneous, block-structured settings by introducing a finite partition of vertices into blocks and a block-aware edge-triangle Hamiltonian. It develops a colored graphon framework and proves a large deviation principle for the associated measures, yielding a variational formula for the limiting free energy. In the ferromagnetic regime where triangle weights are nonnegative, the variational problem collapses to a finite-dimensional scalar fixed-point system for the block-connection matrix, providing a block-wise replica-symmetric analogue. Under a Dobrushin-type condition, the maximizer is unique and a strong law of large numbers holds for the block-edge density, establishing precise asymptotic behavior and a clear route to studying symmetry-breaking phenomena in more general regimes.

Abstract

We extend the classical edge-triangle Exponential Random Graph Model (ERGM) to an inhomogeneous setting in which vertices carry types determined by an underlying partition. This leads to a block-structured ERGM where interaction parameters depend on vertex types. We establish a large deviation principle for the associated sequence of measures and derive the corresponding variational formula for the limiting free energy. In the ferromagnetic regime, where the parameters governing triangle densities are nonnegative, we reduce the variational problem to a scalar optimization problem, thereby identifying the natural block counterpart of the replica symmetric regime. Under additional restrictions on the parameters, comparable to the classical Dobrushin's uniqueness region, we prove uniqueness of the maximizer and derive a law of large numbers for the edge density.
Paper Structure (17 sections, 12 theorems, 160 equations, 3 figures)

This paper contains 17 sections, 12 theorems, 160 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Structure
  4. Preliminaries
  5. The model
  6. The space of colored graphons
  7. Main results
  8. Proofs
  9. LDP
  10. Euler-Lagrange equations
  11. Scalar problem for the free energy
  12. Uniqueness
  13. SLLN
  14. Other helpful lemmas
  15. Erdős-Rényi LDP
  16. ...and 2 more sections

Key Result

Lemma 2.8

The space $(\widetilde{W}_{\mathcal{B}}^{(k)},\delta^{(k)}_{\square})$ is compact.

Figures (3)

  • Figure 2.1: Graph $H$ with $m=6$ vertices endowed with a partition into $k=3$ colors (left), and the associated colored checkerboard graphon $g_{\mathcal{B}^{(3)}_n}^{H}$ (right).
  • Figure 2.2: Example of a partition into equivalence classes (light-blue shaded) when $k=2$ and $n=3$.
  • Figure 2.3: Illustration of a maximizer $(g_{C^{\star}},\mathcal{B}^{k})$ of \ref{['varpr']}, where $C^{\star}$ maximizes the scalar problem \ref{['eq:finite-dim-problem']} and satisfies the system in \ref{['eq:fixed-point-scalar']}.

Theorems & Definitions (46)

  • Remark 2.2: Notation
  • Remark 2.3: Reduction to Erdős-Rényi model
  • Remark 2.4
  • Definition 2.5: Checkerboard graphon
  • Definition 2.6
  • Definition 2.7: Equivalence relation
  • Lemma 2.8: Compactness
  • proof
  • Definition 2.9: Cell-restricted subgraph density
  • Definition 2.10: Entropy and interaction functionals
  • ...and 36 more