A large deviation principle for the Erd\H{o}s-R\'enyi uniform random graph

TL;DR

This paper establishes a large deviation principle for uniform random graphs with a fixed number of edges, extending previous results for Erdős-Rényi graphs and linking tail behaviors of subgraph counts to variational problems.

Contribution

It derives the LDP for uniform random graphs with fixed edge count, connecting tail decay analysis to variational problems similar to those in constrained random graph models.

Findings

LDP for uniform random graphs with fixed edges derived

Tail decay of subgraph counts characterized by variational problems

Connections made to constrained random graph models like edge/triangle

Abstract

Starting with the large deviation principle (LDP) for the Erd\H{o}s-R\'enyi binomial random graph $\mathcal{G}(n,p)$ (edge indicators are i.i.d.), due to Chatterjee and Varadhan (2011), we derive the LDP for the uniform random graph $\mathcal{G}(n,m)$ (the uniform distribution over graphs with $n$ vertices and $m$ edges), at suitable $m=m_n$. Applying the latter LDP we find that tail decays for subgraph counts in $\mathcal{G}(n,m_n)$ are controlled by variational problems, which up to a constant shift, coincide with those studied by Kenyon et al. and Radin et al. in the context of constrained random graphs, e.g., the edge/triangle model.