Electronic Journal of Probability

Ensemble equivalence for dense graphs

F. den Hollander, M. Mandjes, A. Roccaverde, and N.J. Starreveld

Full-text: Open access


In this paper we consider a random graph on which topological restrictions are imposed, such as constraints on the total number of edges, wedges, and triangles. We work in the dense regime, in which the number of edges per vertex scales proportionally to the number of vertices $n$. Our goal is to compare the micro-canonical ensemble (in which the constraints are satisfied for every realisation of the graph) with the canonical ensemble (in which the constraints are satisfied on average), both subject to maximal entropy. We compute the relative entropy of the two ensembles in the limit as $n$ grows large, where two ensembles are said to be equivalent in the dense regime if this relative entropy divided by $n^2$ tends to zero. Our main result, whose proof relies on large deviation theory for graphons, is that breaking of ensemble equivalence occurs when the constraints are frustrated. Examples are provided for three different choices of constraints.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 12, 26 pp.

Received: 27 March 2017
Accepted: 3 January 2018
First available in Project Euclid: 12 February 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

random graph canonical ensemble microcanonical ensemble constraint ensemble equivalence relative entropy graphon variational representation

Creative Commons Attribution 4.0 International License.


den Hollander, F.; Mandjes, M.; Roccaverde, A.; Starreveld, N.J. Ensemble equivalence for dense graphs. Electron. J. Probab. 23 (2018), paper no. 12, 26 pp. doi:10.1214/18-EJP135. https://projecteuclid.org/euclid.ejp/1518426060

Export citation


  • [1] D. Aristoff and L. Zhu, Asymptotic structure and singularities in constrained directed graphs, Stoch. Proc. Appl. 125 (2011) 4154–4177.
  • [2] D. Aristoff and L. Zhu, On the phase transition curve in a directed exponential random graph model, arXiv:1404.6514.
  • [3] S. Bhamidi, G. Bresler and A. Sly, Mixing time of exponential random graphs, Ann. Appl. Probab. 21 (2011) 2146–2170.
  • [4] C. Borgs, J.T. Chayes, L. Lovász, V.T. Sós and K. Vesztergombi, Convergent graph sequences I: Subgraph frequencies, metric properties, and testing, Adv. Math. 219 (2008) 1801–1851.
  • [5] C. Borgs, J.T. Chayes, L. Lovász, V.T. Sós and K. Vesztergombi, Convergent sequences of dense graphs II: Multiway cuts and statistical physics, Ann. Math. 176 (2012) 151–219.
  • [6] S. Chatterjee, An introduction to large deviations for random graphs, Bull. Amer. Math. Soc. 53 (2016) 617–642.
  • [7] S. Chatterjee, Large deviations for Random Graphs, École d’ Été de Probabilités de Saint-Flour XLV, Springer Lecture Notes in Mathematics, 2015.
  • [8] S. Chatterjee and A. Dembo, Non linear large deviations, Adv. Math. 299 (2016) 396–450.
  • [9] S. Chatterjee and P. Diaconis, Estimating and understanding exponential random graph models, Ann. Stat. 41 (2013) 2428–2461.
  • [10] S. Chatterjee, P. Diaconis and A. Sly, Random graphs with a given degree sequence, Ann. Appl. Probab. 21 (2011) 1400–1435.
  • [11] S. Chatterjee and S.R.S. Varadhan, The large deviation principle for the Erdős-Rényi random graph, European J. Comb. 32 (2011) 1000–1017.
  • [12] P. Diao, D. Guillot, A. Khare and B. Rajaratnam, Differential calculus on graphon space, J. Combin. Theory Ser. A 133 (2015) 183–227.
  • [13] P. Erdős, D.J. Kleitman and B.L. Rothschild, Asymptotic enumeration of $K_n$-free graphs, Colloquio Internationale sulle Teorie Combinatorie, 1973, Rome.
  • [14] J.W. Gibbs, Elementary Principles of Statistical Mechanics, Yale University Press, New Haven, Connecticut, 1902.
  • [15] G. Garlaschelli, F. den Hollander and A. Roccaverde, Ensemble equivalence in random graphs with modular structure, J. Phys. A: Math. Theor. 50 (2017).
  • [16] R. van der Hofstad, Random Graphs and Complex Networks, Volume I, Cambridge University Press, Cambridge, 2017.
  • [17] E.T. Jaynes, Information theory and statistical mechanics, Phys. Rev. 106 (1957) 620–630.
  • [18] R. Kenyon, C. Radin, K. Ren and L. Sadun, Multipodal structure of phase transitions in large constrained graphs, J. Stat. Phys. 168 (2017) 233–258.
  • [19] R. Kenyon and M. Yin, On the asymptotics of constrained exponential random graphs, J. Appl. Prob. 54 (2017) 165–180.
  • [20] L. Lovász and B. Szegedy, Limits of dense graph sequences, J. Combin. Theory Ser. B 96 (2006) 933–957.
  • [21] E. Lubetzky and Y. Zhao, On replica symmetry of large deviations in random graphs, Random Structures and Algorithms 47 (2015) 109–146.
  • [22] J. Park and M.E.J. Newman, Statistical mechanics of networks, Phys. Rev. E 70 (2014) 066117.
  • [23] C. Radin and L. Sadun, Phase transitions in a complex network, J. Phys. A: Math. Theor. 46 (2013) 305002.
  • [24] C. Radin and L. Sadun, Singularities in the entropy of asymptotically large simple graphs, J. Stat. Phys. 158 (2015) 853–865.
  • [25] C. Radin and M. Yin, Phase transitions in exponential random graphs, Ann. Appl. Probab. 23 (2013) 2458–2471.
  • [26] O. Pikhurko and A. Razborov, Asymptotic structure of graphs with the minimum number of triangles, Comb. Prob. Comp. 26 (2017) 138–160.
  • [27] T. Squartini, J. de Mol, F. den Hollander and D. Garlaschelli, Phys. Rev. Lett. 115 (2015) 268701.
  • [28] H. Touchette, General equivalence and nonequivalence of ensembles: Thermodynamic, macrostate, and measure levels, J. Stat. Phys. 159 (2015) 987–1016.
  • [29] M. Yin, Critical Phenomena in Exponential Random Graphs, J. Stat. Phys. 153 (2013) 1008–1021.
  • [30] M. Yin, Large deviations and exact asymptotics for constrained exponential random graphs, Electron. Commun. Probab. 20 (2015) 14 pp.
  • [31] M. Yin and L. Zhu, Asymptotics for sparse exponential random graph models, Braz. J. Probab. Stat. 31 (2017) 394–412.
  • [32] L. Zhu, Asymptotic Structure of constrained exponential random graph models, J. Stat. Phys. 166 (2017) 1464–1482.