Exponential bounds for inhomogeneous random graphs in a Gaussian case

Rank 1 inhomogeneous random graphs are a natural generalization of Erd\H{o}s R\'enyi random graphs. In this generalization each node is given a weight. Then the probability that an edge is present depends on the product of the weights of the nodes it is connecting. In this article, we give precise and uniform exponential bounds on the size, weight and surplus of rank 1 inhomogeneous random graphs where the weights of the nodes behave like a random variable with finite fourth moment. We focus on the case where the mean degree of a random node is slightly larger than 1, we call that case the barely supercritical regime. These bounds will be used in follow up articles to study a general class of random minimum spanning trees. They are also of independent interest since they show that these inhomogeneous random graphs behave like Erd\H{o}s R\'enyi random graphs even in a barely supercritical regime. The proof relies on novel concentration bounds for sampling without replacement and a careful study of the exploration process.