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On the asymptotics of constrained exponential random graphs

Part of: Graph theory Equilibrium statistical mechanics

Published online by Cambridge University Press:  04 April 2017

Richard Kenyon  and
Mei Yin
Richard Kenyon*
Affiliation:
Brown University
Mei Yin*
Affiliation:
University of Denver
*
* Postal address: Department of Mathematics, Brown University, Providence, RI 02912, USA. Email address: rkenyon@math.brown.edu
** Postal address: Department of Mathematics, University of Denver, Denver, CO 80208, USA. Email address: mei.yin@du.edu
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Abstract

The unconstrained exponential family of random graphs assumes no prior knowledge of the graph before sampling, but it is natural to consider situations where partial information about the graph is known, for example the total number of edges. What does a typical random graph look like, if drawn from an exponential model subject to such constraints? Will there be a similar phase transition phenomenon (as one varies the parameters) as that which occurs in the unconstrained exponential model? We present some general results for this constrained model and then apply them to obtain concrete answers in the edge-triangle model with fixed density of edges.

Keywords

Constrained exponential random graphphase transition

MSC classification

Primary: 05C80: Random graphs
Secondary: 82B26: Phase transitions (general)
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 1 , March 2017 , pp. 165 - 180
Copyright
Copyright © Applied Probability Trust 2017 

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