The asymptotics of large constrained graphs

We show, through local estimates and simulation, that if one constrains simple graphs by their densities of edges and τ of triangles, then asymptotically (in the number of vertices) for over 95 % of the possible range of those densities there is a well-defined typical graph, and it has a very simple structure: the vertices are decomposed into two subsets V1 and V2 of fixed relative size c and 1 − c, and there are well-defined probabilities of edges, gjk, between vj ∈ Vj, and vk ∈ Vk. Furthermore the four parameters c, g11, g22 and g12 are smooth functions of (, τ) except at two smooth ‘phase transition ’ curves.