An introduction to large deviations for random graphs

TL;DR

This paper provides a comprehensive overview of large deviation principles in random graph theory, covering dense and sparse regimes, key techniques, and recent advances for a general mathematical audience.

Contribution

It synthesizes recent progress in large deviations for random graphs, including dense and sparse cases, and discusses new methods beyond traditional tools like the regularity lemma.

Findings

Large deviation principles are established for dense random graphs using graph limit theory.

New techniques have been developed for sparse graphs where regularity lemma does not apply.

Applications to exponential random graph models are briefly discussed.

Abstract

This article gives an overview of the emerging literature on large deviations for random graphs. Written for the general mathematical audience, the article begins with a short introduction to the theory of large deviations. This is followed by a description of some large deviation questions about random graphs, and an outline of the recent progress on this topic. A more elaborate discussion follows, with a brief account of graph limit theory and its application in constructing a large deviation theory for dense random graphs. The role of Szemer\'edi's regularity lemma is explained, together with a sketch of the proof of the main large deviation result and some examples. Applications to exponential random graph models are briefly touched upon. The remainder of the paper is devoted to large deviations for sparse graphs. Since the regularity lemma is not applicable in the sparse regime, new tools are needed. Fortunately, there have been several new breakthroughs that managed to achieve the goal by an indirect method. These are discussed, together with an exposition of the underlying theory. The last section contains a list of open problems.