The asymptotics of large constrained graphs

Charles Radin, Kui Ren, Lorenzo Sadun
2014 Journal of Physics A: Mathematical and Theoretical  
We show, through local estimates and simulation, that if one constrains simple graphs by their densities ε 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 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∈ V_j, and v_k∈ V_k.
more » ... more the four parameters c, g_11, g_22 and g_12 are smooth functions of (ε,τ) except at two smooth 'phase transition' curves.
doi:10.1088/1751-8113/47/17/175001 fatcat:jx6ym4kzzfb2zezllv3n5zcb4q