  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 $H$.
