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

Part of: Equilibrium statistical mechanics Graph theory

Published online by Cambridge University Press:  20 March 2018

David Aristoff  and
Lingjiong Zhu
David Aristoff*
Affiliation:
Colorado State University
Lingjiong Zhu*
Affiliation:
Florida State University
*
* Postal address: Department of Mathematics, Colorado State University, 1874 Campus Delivery, Fort Collins, CO-80523, USA. Email address: aristoff@math.colostate.edu
** Postal address: Department of Mathematics, Florida State University, 1017 Academic Way, Tallahassee, FL-32306, USA. Email address: zhu@math.fsu.edu
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Abstract

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.

Keywords

Dense random graphexponential random graphgraph limitentropyphase transition

MSC classification

Primary: 05C80: Random graphs 82B26: Phase transitions (general)
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 1 , March 2018 , pp. 272 - 301
Copyright
Copyright © Applied Probability Trust 2018 

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