Extremal Behavior in Exponential Random Graphs

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.