An introduction to large deviations for random graphs

Abstract

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 Szemerédi'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.

Nonlinear large deviations

“…The above result has been recently extended to more general subgraph counts by Bhattacharya et al [2]. For a survey of these developments and a short overview of the emerging field of large deviations for random graphs, see [13]. We want to investigate conditions under which the following "upper tail approximation" is valid: P(f (Y ) ≥ tn) = exp(−φ p (t) + lower order terms) .…”

“…Unlike the dense case, it is hard to give a precise meaning to claims about the conditional structure in the sparse setting due to the lack of an adequate sparse graph limit theory. For a detailed discussion, see [2,13].…”

Upper Tail Large Deviations for Arithmetic Progressions in a Random Set

“…YZ was supported by an Esmée Fairbairn Junior Research Fellowship at New College, Oxford and NSF Award DMS-1362326. an Erdős-Rényi random graph G(N, p). See Chatterjee's recent survey [7] and the references therein for an introduction to recent developments on large deviations in random graphs.…”

Bivariate fluctuations for the number of arithmetic progressions in random sets

“…the upper tail probabilities P (X ℓ ≥ (1 + ε)E(X ℓ )), and the lower tail probabilities P (X ℓ ≤ (1 − ε)E(X ℓ )). For a recent survey on large deviations in random graphs (and related combinatorial structures) see [7].…”