Moderate deviations of subgraph counts in the Erd\H{o}s-R\'enyi random graphs $G(n,m)$ and $G(n,p)$

TL;DR

This paper derives asymptotic expressions for the probabilities of moderate deviations in subgraph counts within Erdős-Rényi random graphs, linking deviations across different subgraphs and extending bounds to the $G(n,p)$ model.

Contribution

It provides the first asymptotic rate for moderate deviations of subgraph counts in $G(n,m)$ and connects deviations of different subgraphs through specific graph structures.

Findings

Asymptotic expressions for deviation rates in $G(n,m)$

Linkage of subgraph deviations via paths and triangles

New bounds for $G(n,p)$ model

Abstract

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.