  Yin, Rinaldo, and Fadnavis classified the extremal behavior of the
edge-triangle exponential random graph model by first taking the network size
to infinity, then the parameters diverging to infinity along straight lines.
Lubetzky and Zhao proposed an extension to the edge-triangle model by
introducing an exponent $\gamma > 0$ on the triangle homomorphism density
function. This allows non-trivial behavior in the positive limit, which is
absent in the standard edge-triangle model. The present work seeks to classify
the limiting behavior of this generalized edge-triangle exponential random
graph model. It is shown that for $\gamma \le 1$, the limiting set of graphons
come from a special class, known as Tur\'an graphons. For $\gamma > 1$, there
are large regions of the parameter space where the limit is not a Tur\'an
graphon, but rather has edge density between subsequent Tur\'an graphons.
Furthermore, for $\gamma$ large enough, the exact edge density of the limiting
set is determined in terms of a nested radical. Utilizing a result of Reiher,
intuition is given for the characterization of the extremal behavior in the
generalized edge-clique model.
