Tame sparse exponential random graphs

In this paper, we obtain a precise estimate of the probability that the sparse binomial random graph contains a large number of vertices in a triangle. The estimate of log of this probability is correct up to second order, and enables us to propose an exponential random graph model based on the number of vertices in a triangle. Specifically, by tuning a single parameter, we can with high probability induce any given fraction of vertices in a triangle. Moreover, in the proposed exponential random graph model we derive the large deviation principle for the number of edges. As a byproduct, we propose a consistent estimator of the tuning parameter.