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On the phase transition curve in a directed exponential random graph model

  • David Aristoff (a1) and Lingjiong Zhu (a2)


We consider a family of directed exponential random graph models parametrized by edges and outward stars. Much of the important statistical content of such models is given by the normalization constant of the models, and, in particular, an appropriately scaled limit of the normalization, which is called the free energy. We derive precise asymptotics for the normalization constant for finite graphs. We use this to derive a formula for the free energy. The limit is analytic everywhere except along a curve corresponding to a first-order phase transition. We examine unusual behavior of the model along the phase transition curve.


Corresponding author

* Postal address: Department of Mathematics, Colorado State University, 1874 Campus Delivery, Fort Collins, CO-80523, USA. Email address:
** Postal address: Department of Mathematics, Florida State University, 1017 Academic Way, Tallahassee, FL-32306, USA. Email address:


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