Asymptotic structure and singularities in constrained directed graphs
  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. 
David Aristoff and Lingjiong Zhu
math.PR math-ph math.CO math.MP