ERGMs on block models
Authors
- Magnanini, Elena
ORCID: 0000-0001-5430-4884
2020 Mathematics Subject Classification
- 05C80 60F15 82B05
Keywords
- Edge-triangle model, free energy, block models, Euler--Lagrange equations, limit theorems
DOI
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
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