Skip to main content
SpringerLink
Log in

Singularities in the Entropy of Asymptotically Large Simple Graphs

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We prove that the asymptotic entropy of large simple graphs, as a function of fixed edge and triangle densities, is nondifferentiable along a certain curve. We also determine the precise modified-bipartite structure of asymptotic graphs with edge density 1/2 and triangle density in the interval \(\displaystyle [0,1/8]\).

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Aristoff, D., Radin, C.: Emergent structures in large networks. J. Appl. Probab. 50, 883–888 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bollobás, B.: Extremal Graph Theory. Academic Press, London (1978)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Lovász, L.: Large networks and graph limits. American Mathematical Society, Providence (2012)

    MATH  Google Scholar 

  8. Lovász, L., Szegedy, B.: Limits of dense graph sequences. J. Combin. Theory Ser. B 96, 933–957 (2006)

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

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  12. O. Pikhurko and A. Razborov, Asymptotic structure of graphs with the minimum number of triangles, arXiv:1203.4393.

  13. Razborov, A.: On the minimal density of triangles in graphs. Combin. Probab. Comput. 17, 603–618 (2008)

    MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  15. Radin, C., Yin, M.: Phase transitions in exponential random graphs. Ann. Appl. Probab. 23, 2458–2471 (2013)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

We gratefully acknowledge useful discussions with Mei Yin. This work was partially supported by NSF Grants DMS-1208941 and DMS-1101326.

Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Texas at Austin, Austin, TX, 78712, USA

    Charles Radin & Lorenzo Sadun

Authors
  1. Charles Radin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lorenzo Sadun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Charles Radin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Radin, C., Sadun, L. Singularities in the Entropy of Asymptotically Large Simple Graphs. J Stat Phys 158, 853–865 (2015). https://doi.org/10.1007/s10955-014-1151-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-014-1151-3

Keywords

Mathematics Subject Classification

Search

Navigation