  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.
