ERGMs on block models

TL;DR

This paper extends ERGMs to block-structured models with vertex types, establishing large deviation principles and analyzing the limiting behavior of these models, including phase transition regimes.

Contribution

It introduces a block-structured ERGM framework, derives large deviation principles, and characterizes the replica symmetric regime and phase transitions.

Findings

Established a large deviation principle for block ERGMs.

Reduced the variational problem to a scalar optimization in the ferromagnetic regime.

Proved uniqueness of the maximizer and a law of large numbers under certain conditions.

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.