Multipodal Structure and Phase Transitions in Large Constrained Graphs

Abstract

We study the asymptotics of large, simple, labeled graphs constrained by the densities of two subgraphs. It was recently conjectured that for all feasible values of the densities most such graphs have a simple structure. Here we prove this in the special case where the densities are those of edges and of k-star subgraphs, $k\ge 2$ fixed. We prove that under such constraints graphs are “multipodal”: asymptotically in the number of vertices there is a partition of the vertices into $M < \infty$ subsets $V_1, V_2, \ldots , V_M$, and a set of well-defined probabilities $g_{ij}$ of an edge between any $v_i \in V_i$ and $v_j \in V_j$. For $2\le k\le 30$ we determine the phase space: the combinations of edge and k-star densities achievable asymptotically. 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.