Phase Transitions in Finite Random Networks


We analyze ensembles of random networks with fixed numbers of edges, triangles, and nodes. In the limit as the number of nodes goes to infinity, this model is known to exhibit sharp phase transitions as the density of edges and triangles is varied. In this paper we study finite size effects in ensembles where the number of nodes is between 30 and 66. Despite the small number of nodes, the phases and phase transitions are clearly visible. Concentrating on 54 nodes and one particular phase transition, we use spectral data to examine the structure of a typical graph in each phase, and see that is it very similar to the graphon that describes the system as n diverges. We further investigate how graphs change their structure from that of one phase to the other under a natural edge flip dynamics in which the number of triangles is either forced downwards or allowed to drift upwards. In each direction, spectral data shows that the change occurs in three stages: a very fast stage of about 100 edge flips, in which the triangle count reaches the targeted value, a slower second stage of several thousand edge flips in which the graph adopts the qualitative features of the new phase, and a very slow third stage, requiring hundreds of thousands of edge flips, in which the fine details of the structure equilibrate.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17


  1. 1.

    Abbe, E.: Community detection and stochastic block models: recent developments. J. Mach. Learn. Res. 18, 6446–6531 (2017)

    MathSciNet  Google Scholar 

  2. 2.

    Borgs, C., Chayes, J., Lovász, L.: Moments of two-variable functions and the uniqueness of graph limits. Geom. Funct. Anal. 19, 1597–1619 (2010)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Borgs, C., Chayes, J., Lovász, L., Sós, V.T., Vesztergombi, K.: Convergent graph sequences I: subgraph frequencies, metric properties, and testing. Adv. Math. 219, 1801–1851 (2008)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Chatterjee, S., Diaconis, P.: Estimating and understanding exponential random graph models. Ann. Stat. 41, 2428–2461 (2013)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Chatterjee, S., Varadhan, S.R.S.: The large deviation principle for the Erdős–Rényi random graph. Eur. J. Comb. 32, 1000–1017 (2011)

    Article  Google Scholar 

  6. 6.

    Kenyon, R., Radin, C., Ren, K., Sadun, L.: Multipodal structure and phase transitions in large constrained graphs. J. Stat. Phys. 168, 233–258 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  7. 7.

    Kenyon, R., Radin, C., Ren, K., Sadun, L.: Bipodal structure in oversaturated random graphs. Int. Math. Res. Notices 2016, 1009–1044 (2018)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Kenyon, R., Radin, C., Ren, K., Sadun, L.: The phases of large networks with edge and triangle constraints. J. Phys. A 50, 435001 (2017)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Koch, H.: Vertex order in some large constrained random graphs. SIAM J. Math. Anal. 48, 2588–2601 (2016)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Lovász, L.: Large Networks and Graph Limits. American Mathematical Society, Providence (2012)

    Google Scholar 

  11. 11.

    Lovász, L., Szegedy, B.: Limits of dense graph sequences. J. Comb. Theory Ser. B 98, 933–957 (2006)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Lovász, L., Szegedy, B.: Szemerédi’s lemma for the analyst. GAFA 17, 252–270 (2007)

    MATH  Google Scholar 

  13. 13.

    Lovász, L., Szegedy, B.: Finitely forcible graphons. J. Comb. Theory Ser. B 101, 269–301 (2011)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Newman, M.E.J.: Networks: An Introduction. Oxford University Press, Oxford (2010)

    Google Scholar 

  15. 15.

    Radin, C., Sadun, L.: Phase transitions in a complex network. J. Phys. A 46, 305002 (2013)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Radin, C., Sadun, L.: Singularities in the entropy of asymptotically large simple graphs. J. Stat. Phys. 158, 853–865 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  17. 17.

    Radin, C., Ren, K., Sadun, L.: The asymptotics of large constrained graphs. J. Phys. A 47, 175001 (2014)

    ADS  MathSciNet  Article  Google Scholar 

  18. 18.

    Radin, C., Ren, K., Sadun, L.: A symmetry breaking transition in the edge/triangle network model. Ann. Inst. H. Poincaré D 5, 251–286 (2018)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Radin, C., Swinney, H.: Phases of granular matter. J. Stat. Phys. 175, 542–553 (2019)

    ADS  Article  Google Scholar 

  20. 20.

    Rietz, F., Radin, C., Swinney, H., Schröter, M.: Nucleation in sheared granular matter. Phys. Rev. Lett. 120, 055701 (2018)

    ADS  Article  Google Scholar 

  21. 21.

    Strauss, D.: On a general class of models for interaction. SIAM Rev. 28, 513–527 (1986)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Uhlenbeck, G.E.: In: Cohen, E.G.D. (ed.) Fundamental Problems in Statistical Mechanics II, pp. 16–17. Wiley, New York (1968)

Download references

Author information



Corresponding author

Correspondence to Joe Neeman.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Federico Ricci-Tersenghi.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Neeman, J., Radin, C. & Sadun, L. Phase Transitions in Finite Random Networks. J Stat Phys (2020).

Download citation


  • Random graph
  • Phase transition
  • Finite size effect
  • Markov Chain Monte Carlo
  • Graphon