Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Low eigenvalues of Laplacian matrices of large random graphs
Download PDF
Download PDF
  • Published: 20 March 2011

Low eigenvalues of Laplacian matrices of large random graphs

  • Tiefeng Jiang1 

Probability Theory and Related Fields volume 153, pages 671–690 (2012)Cite this article

  • 382 Accesses

  • 8 Citations

  • Metrics details

Abstract

For each n ≥ 2, let A n  = (ξ ij ) be an n × n symmetric matrix with diagonal entries equal to zero and the entries in the upper triangular part being independent with mean μ n and standard deviation σ n . The Laplacian matrix is defined by \({{\bf \Delta}_n={\rm diag}(\sum_{j=1}^n\xi_{ij})_{1\leq i \leq n}-{\bf A}_n}\). In this paper, we obtain the laws of large numbers for λ n–k (Δ n ), the (k + 1)-th smallest eigenvalue of Δ n , through the study of the order statistics of weakly dependent random variables. Under certain moment conditions on ξ ij ’s, we prove that, as n → ∞,

$$({\rm i})\quad\frac{\lambda_{n-k}({\bf \Delta}_n)-n\mu_n} {\sigma_n\sqrt{n\log n}} \to -\sqrt{2} \quad a.s. $$

for any k ≥ 1. Further, if {Δ n ; n ≥ 2} are independent with μ n  = 0 and σ n  = 1, then, (ii) the sequence \({\;\left\{\frac{\lambda_{n-k}({\bf \Delta}_n)}{\sqrt{n\log n}};n\geq 2\right\}}\) is dense in \({\left[-\sqrt{2+2(k+1)^{-1}}, -\sqrt{2}\,\right]\ a.s.}\) for any k ≥ 0. In particular, (i) holds for the Erdös–Rényi random graphs. Similar results are also obtained for the largest eigenvalues of Δ n .

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Bahatia, R.: Matrix analysis. In: Graduate Texts in Mathematics, vol. 169. Springer, Berlin (1997)

  2. Bai Z.: Methodologies in spectral analysis of large dimensional random matrices: a review. Stat. Sin. 9, 611–677 (1999)

    MATH  Google Scholar 

  3. Bauer M., Golinelli O.: Random incidence matrices: moments of the spectral density. J. Stat. Phys. 103, 301–337 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bollobás B.: Random Graphs. Academic Press, New York (1985)

    MATH  Google Scholar 

  5. Bordenave C., Caputo P., Chafaï D.: Spectrum of large random reversible Markov chains: two examples. Latin Am. J. Probab. Math. Stat. 7, 41–64 (2010)

    Google Scholar 

  6. Bryc W., Dembo A., Jiang T.: Spectral measure of large random Hankel, Markov and Toeplitz matrices. Ann. Probab. 34, 1–38 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chung, F.: Spectral graph theory. In: CBMS Regional Conference Series in Mathematics, vol. 92. American Mathematical Society (1997)

  8. Chung, F., Lu, L.: Complex graphs and networks. In: CBMS Regional Conference Series in Mathematics, vol. 107. American Mathematical Society (2006)

  9. Colin de Verdière, Y.: Spectres de Graphes. Societe Mathematique De France (1998)

  10. David, H., Nagaraja, H.: Order Statistics, 3rd edn. Wiley-Interscience (2003)

  11. Ding X., Jiang T.: Spectral distributions of adjacency and Laplacian matrices of weighted random graphs. Ann. Appl. Probab. 20(6), 2086–2117 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Durrett, R.: Random Graph Dynamics. Cambridge University Press (2006)

  13. Erdös P., Rényi A.: On the evolution of random graphs. Magyar Tud. Akad. Mat. Kuató Int. Közl 5, 17–61 (1960)

    MATH  Google Scholar 

  14. Erdös P., Rényi A.: On random graphs I. Publ. Math. Debrecen. 6, 290–297 (1959)

    MathSciNet  MATH  Google Scholar 

  15. Horn, R., Johnson, C.: Matrix Analysis. Cambridge Univesity Press (1985)

  16. Fiedler M.: Algebraic connectivity of graphs (English). Czechoslov. Math. J. 23(2), 298–305 (1973)

    MathSciNet  Google Scholar 

  17. Janson S., Łuczak T., Ruciński A.: Random Graphs. Wiley, New York (2000)

    Book  MATH  Google Scholar 

  18. Khorunzhy O., Shcherbina M., Vengerovsky V.: Eigenvalue distribution of large weighted random graphs. J. Math. Phys. 45(4), 1648–1672 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Khorunzhy A., Khoruzhenko B., Pastur L., Shcherbina M.: The Large n-limit in Statistical Mechanics and the Spectral Theory of Disordered Systems, Phase Transition and Critical Phenomenon, vol. 15, pp. 73. Academic, New York (1992)

    Google Scholar 

  20. Kirchhoff G.: Über die Auflösung der Gleichungen, auf welche man bei der untersuchung der linearen verteilung galvanischer Ströme geföhrt wird. Ann. Phys. Chem. 72, 497–508 (1847)

    Google Scholar 

  21. Kolchin V.: Random Graphs. Cambridge University Press, New York (1998)

    Book  Google Scholar 

  22. Lin Z., Bai Z.: Probability Inequalities, 1st edn. Springer, Berlin (2011)

    Book  Google Scholar 

  23. Mohar, B.: In: Alavi, Y., Chartrand, G., Oellermann, O.R., Schwenk, A.J. (eds.) Graph Theory, Combinatorics, and Applications, vol. 2, pp. 871–898. Wiley, New York (1991)

  24. Palmer E.: Graphical Evolution: An Introduction to the Theory of Random Graphs. Wiley, New York (1985)

    MATH  Google Scholar 

  25. Rogers G., De Dominicis C.: Density of states of sparse random matrices. J. Phys. A: Math. Gen. 23, 1567–1573 (1990)

    Article  Google Scholar 

  26. Rogers G., Bray A.: Density of states of a sparse random matrix. Phys. Rev. B. 37, 3557–3562 (1988)

    Article  MathSciNet  Google Scholar 

  27. Sakhanenko, A.: On the accuracy of normal approximation in the invariance principle [translation of Trudy Inst. Mat. (Novosibirsk) 13 (1989), Asimptot. Analiz Raspred. Sluch. Protsess., 40–66; MR 91d:60082]. Sib. Adv. Math. 1(4), 58–91 (1991)

  28. Sakhanenko, A.: Estimates in an invariance principle. In: Limit Theorems of Probability Theory. Trudy Inst. Mat., vol. 5, pp. 27–44, 175. “Nauka” Sibirsk. Otdel, Novosibirsk (1985)

Download references

Author information

Authors and Affiliations

  1. School of Statistics, University of Minnesota, 313 Ford Hall, 224 Church Street S.E., Minneapolis, MN, 55455, USA

    Tiefeng Jiang

Authors
  1. Tiefeng Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tiefeng Jiang.

Additional information

T. Jiang was supported in part by NSF #DMS-0449365.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Jiang, T. Low eigenvalues of Laplacian matrices of large random graphs. Probab. Theory Relat. Fields 153, 671–690 (2012). https://doi.org/10.1007/s00440-011-0357-4

Download citation

  • Received: 29 December 2009

  • Revised: 02 March 2011

  • Published: 20 March 2011

  • Issue Date: August 2012

  • DOI: https://doi.org/10.1007/s00440-011-0357-4

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Random graph
  • Random matrix
  • Laplacian matrix
  • Extreme eigenvalues
  • Order statistics

Mathematics Subject Classification (2000)

  • 05C80
  • 05C50
  • 15A52
  • 60B10
  • 62G30
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature