Multipodal Structure and Phase Transitions in Large Constrained Graphs

We study the asymptotics of large, simple, labeled graphs constrained by the densities of edges and of k-star subgraphs, k ≥ 2 fixed. We prove that under such constraints graphs are “multipodal”: asymptotically in the number of vertices there is a partition of the vertices into M <∞ subsets V1, V2, . . . , VM , and a set of well-defined probabilities gij of an edge between any vi ∈ Vi and vj ∈ Vj . For 2 ≤ k ≤ 30 we determine the phase space: the combinations of edge and k-star densities achievable asymptotically. We also derive the phase space for the triple of densities: edge, 2-star and 3-star. For these models there are special points on the boundary of the phase space with nonunique asymptotic (graphon) structure; for the 2-star model we prove that the nonuniqueness extends to entropy maximizers in the interior of the phase space.