Asymptotic structure and singularities in constrained directed graphs

David Aristoff, Lingjiong Zhu

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


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, in which edges and certain other subgraph densities are controlled by parameters. 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.

Original languageEnglish (US)
Pages (from-to)4154-4177
Number of pages24
JournalStochastic Processes and their Applications
Issue number11
StatePublished - Aug 22 2015


  • Dense random graphs
  • Entropy
  • Exponential random graphs
  • Graph limits
  • Phase transitions


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