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 $95\mathrm{\%}$ 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 $v_j\in V_j$, and $v_k\in V_k$. 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.