For subcritical exponential random graphs conditioned on edge count, the two-star count satisfies a central limit theorem with explicit mean and variance, at rate n^{-1/2+ε} in Wasserstein distance.
Second-order Approximation of Exponential Random Graph Models
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abstract
Exponential random graph models (ERGMs) are flexible probability models allowing edge dependency. However, it is known that, to a first-order approximation, many ERGMs behave like Erd\"os-R\'enyi random graphs, where edges are independent. In this paper, to distinguish ERGMs from Erd\"os-R\'enyi random graphs, we consider second-order approximations of ERGMs using two-stars and triangles. We prove that the second-order approximation indeed achieves second-order accuracy in the triangle-free case. The new approximation is formally obtained by Hoeffding decomposition and rigorously justified using Stein's method.
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Conditional central limit theorems for exponential random graphs
For subcritical exponential random graphs conditioned on edge count, the two-star count satisfies a central limit theorem with explicit mean and variance, at rate n^{-1/2+ε} in Wasserstein distance.