The main contribution of this article is an asymptotic expression for the rate associated with moderate deviations of subgraph counts in the Erd\H{o}s-R\'enyi random graph $G(n,m)$. Our approach is based on applying Freedman's 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.

中文翻译：

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Erdős-Rényi 随机图 $G(n,m)$ 和 $G(n,p)$ 中子图计数的中等偏差

本文的主要贡献是与 Erd\H{o}sR\'enyi 随机图 $G(n,m)$ 中子图计数的中等偏差相关的比率的渐近表达式。我们的方法是基于将弗里德曼的鞅偏差概率的不等式应用于子图计数偏差的鞅表示。此外，我们证明了不同子图的子图计数偏差都是有联系的，通过两个特定图的偏差，长度为2的路径和三角形。我们还为相关的 $G(n,p)$ 模型推导出新的界限。