The large deviation principle for the Erdős-Rényi random graph

Abstract

What does an Erdős-Rényi graph look like when a rare event happens? This paper answers this question when p is fixed and n tends to infinity by establishing a large deviation principle under an appropriate topology. The formulation and proof of the main result uses the recent development of the theory of graph limits by Lovász and coauthors and Szemerédi's regularity lemma from graph theory. As a basic application of the general principle, we work out large deviations for the number of triangles in G(n, p). Surprisingly, even this simple example yields an interesting double phase transition.

The Lower Tail: Poisson Approximation Revisited

“…It frequently serves as a test-bed for new probabilistic estimates (see, e.g., [2,15,27,21,18,17,7]), and we shall use it to demonstrate the applicability of our bootstrapping approaches. In fact, we consider the more general random hypergraph G n is included, independently, with probability p. Given a k-uniform hypergraph H, or briefly k-graph, we define X H = X H (n, p) as the number of copies of H in G (k) n,p , where by a copy we mean, as usual, a subgraph isomorphic to H. Furthermore, we write e H = |E(H)| and v H = |V (H)| for the number of edges and vertices of H, respectively.…”

On the variational problem for upper tails in sparse random graphs

“…In the dense case (p fixed), the limiting asymptotics of the rate function -the normalized logarithm of this probability, here denoted by r(n, p, δ) -was reduced to an analytic variational problem on symmetric functions f : [0, 1] 2 → [0, 1] (for a large class of large deviation questions) by Chatterjee and Varadhan [6]. However, for p = o(1), obtaining the order of r(n, p, δ) was already a longstanding open problem.…”

Asymptotic Structure of Constrained Exponential Random Graph Models

“…As a result of the large deviations for random graphs [6] and Varadhan's lemma from large deviation theory, we have the following result.…”

“…Then, using the large deviation theory for random graphs developed in Chatterjee and Varadhan [6], the limiting free energy for the exponential random graph models was obtained in Chatterjee and Diaconis [5].…”