  We study the asymptotics of large, simple, labeled graphs constrained by the
densities 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.
