Moderate deviations of subgraph counts in the ErdH{o}s-Renyi random graphs $G(n,m)$ and $G(n,p)$

The main contribution of this article is an asymptotic expression for the rate associated with moderate deviations of subgraph counts in the ErdH{o}s-Renyi random graph $G(n,m)$. Our approach is based on applying Freedmans inequalities for the probability of deviations of martingales to a martingale representation of subgraph count deviations. In addition, we prove that subgraph count deviations of different subgraphs are all linked, via the deviations of two specific graphs, the path of length two and the triangle. We also deduce new bounds for the related $G(n,p)$ model.