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.

翻译：本文中，我们获得了稀疏二项随机图中包含大量三角形顶点数的概率的精确估计。该概率对数的估计精确至二阶项，使我们能够提出一种基于三角形顶点数的指数随机图模型。具体而言，通过调节单一参数，我们能够以高概率诱导任意给定比例的三角形顶点数。此外，在所提出的指数随机图模型中，我们推导了边数的大偏差原理。作为副产品，我们提出了该调节参数的一致性估计量。