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We study the asymptotics of large, simple, labeled graphs constrained by thedensities of edges and of $k$-star subgraphs, $k\ge 2$ fixed. We prove thatunder such constraints graphs are "multipodal": asymptotically in the number ofvertices 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 edgebetween any $v_i \in V_i$ and $v_j \in V_j$. For $2\le k\le 30$ we determinethe phase space: the combinations of edge and $k$-star densities achievableasymptotically. For these models there are special points on the boundary ofthe phase space with nonunique asymptotic (graphon) structure; for the 2-starmodel we prove that the nonuniqueness extends to entropy maximizers in theinterior of the phase space.

Multipodal Structure and Phase Transitions in Large Constrained Graphs