  We study the asymptotics of large directed graphs, constrained to have
certain densities of edges and/or outward $p$-stars. Our models are close
cousins of exponential random graph models (ERGMs), in which edges and certain
other subgraph densities are controlled by parameters. The idea of directly
constraining edge and other subgraph densities comes from Radin and Sadun. Such
modeling circumvents a phenomenon first made precise by Chatterjee and
Diaconis: that in ERGMs it is often impossible to independently constrain edge
and other subgraph densities. In all our models, we find that large graphs have
either uniform or bipodal structure. When edge density (resp. $p$-star density)
is fixed and $p$-star density (resp. edge density) is controlled by a
parameter, we find phase transitions corresponding to a change from uniform to
bipodal structure. When both edge and $p$-star density are fixed, we find only
bipodal structures and no phase transition.
