Bipodal structure in oversaturated random graphs

We study the asymptotics of large simple graphs constrained by the limiting density of edges and the limiting subgraph density of an arbitrary fixed graph $H$. We prove that, for all but finitely many values of the edge density, if the density of $H$ is constrained to be slightly higher than that for the corresponding Erd\H{o}s-R\'enyi graph, the typical large graph is bipodal with parameters varying analytically with the densities. Asymptotically, the parameters depend only on the degree sequence of $H$

Existence of a symmetric bipodal phase in the edge-triangle model

In the edge-triangle model with edge density close to 1/2 and triangle density below 1/8 we prove that the unique entropy-maximizing graphon is symmetric bipodal. We also prove that,for any edge density $e$ less than $e_0 = (3-\sqrt{3})/6 \approx 0.2113$ and triangle density slightly less than $e^3$, the entropy-maximizing graphon is not symmetric bipodal.Comment: Large change of Section

Structure of lower tails in sparse random graphs

We study the typical structure of a sparse Erd\H{o}s--R\'enyi random graph conditioned on the lower tail subgraph count event. We show that in certain regimes, a typical graph sampled from the conditional distribution resembles the entropy minimizer of the mean field approximation in the sense of both subgraph counts and cut norm. The main ingredients are an adaptation of an entropy increment scheme of Kozma and Samotij, and a new stability for the solution of the associated entropy variational problem. Our proof suggests a more general framework for establishing typical behavior statements when the objects of interest can be encoded in a hypergraph satisfying mild degree conditions.Comment: 13 pages, comments welcome