On the Lower Tail Variational Problem for Random Graphs

Abstract

Abstract. We study the lower tail large deviation problem for subgraph counts in a random graph. Let XH denote the number of copies of H in an Erdős-Rényi random graph G(n, p). We are interested in estimating the lower tail probability P(XH ≤ (1 − δ)EXH ) for fixed 0 < δ < 1.Thanks to the results of Chatterjee, Dembo, and Varadhan, this large deviation problem has been reduced to a natural variational problem over graphons, at least for p ≥ n −α H (and conjecturally for a larger range of p). We study this variational problem and provide a partial characterization of the so-called "replica symmetric" phase. Informally, our main result says that for every H, and 0 < δ < δH for some δH > 0, as p → 0 slowly, the main contribution to the lower tail probability comes from Erdős-Rényi random graphs with a uniformly tilted edge density. On the other hand, this is false for non-bipartite H and δ close to 1.

Moderate deviations of subgraph counts in the Erdős-Rényi random graphs 𝐺(𝑛,𝑚) and 𝐺(𝑛,𝑝)

“…Interest in these problems grew after the seminal articles of Vu [28] and Janson and Ruciński [15] in the early 2000s provided many results, using a large range of techniques, which were still far from best possible. Important subsequent advances include the translation of such deviation problems into variational problems for graphons (Chatterjee and Varadhan [7]) and solutions to these variational problems for certain values of the parameters (Lubetzky and Zhao [20] and Zhao [30]). We recommend the survey of Chatterjee [6] and the references therein for a more detailed overview.…”

Approximating the Cumulant Generating Function of Triangles in the Erdös–Rényi Random Graph

“…Thus, in the Erdös-Rényi case the entries of the adjacency matrix, x = (x i, j ) i, j∈ [n] , form a set of independent identically distributed (i.i.d.) Bernoulli variables with P(x i, j = 1) = p. In spite of the simple probabilistic set-up the large deviation principle for the Erdös-Rényi graph is far from simple and has been developed only in recent years [12][13][14]16,17,[25][26][27][28]35]. In particular, it has been found that the large deviation function may be non-convex.…”

An introduction to large deviations for random graphs

“…On the other hand, for the number of graphs with at most tn 3 triangles, such an explicit formula can be obtained if t is sufficiently away from zero, and it can also be shown that this formula does not hold if t is sufficiently close to zero. As of now, there is no explicit formula for small t. See Zhao [52] for the most advanced results about the lower tail problem.…”