The asymptotics of large constrained graphs

We show, through local estimates and simulation, that if one constrains simple graphs by their densities epsilon of edges and tau 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 V-1 and V-2 of fixed relative size c and 1 - c, and there are well-defined probabilities of edges, g(jk), between v(j) is an element of V-j, and v(k) is an element of V-k. Furthermore the four parameters c, g(11), g(22) and g(12) are smooth functions of (epsilon, tau) except at two smooth 'phase transition' curves.