Skip to main content
SpringerLink
Log in

Metric and Spectral Properties of Dense Inhomogeneous Random Graphs

  • Conference paper
  • First Online:
Extended Abstracts Summer 2015

Part of the book series: Trends in Mathematics ((RPCRMB,volume 6))

  • 443 Accesses

Abstract

We study metric and spectral properties of dense inhomogeneous random graphs. We generalize results known for the Erdös–Renyi model. In our case an edge (i, j) is present with probability κ(X i , X j )p, where κ ≥ 0 is a fixed kernel and X i are independent variables from a general distribution on a separable metric space.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 149.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. L. Addario-Berry, N. Broutin, and C. Goldschmidt, “The continuum limit of critical random graphs”, Probability Theory and Related Fields 152 (3–4) (2012), 367–406.

    Google Scholar 

  2. B. Bollobás, “The diameter of random graphs”, Transactions of the American Mathematical Society 267 (1) (1981), 41–52.

    Google Scholar 

  3. B. Bollobás, S. Janson, and O. Riordan, “The phase transition in inhomogeneous random graphs”, Random Structures & Algorithms 31 (2007), 3–122.

    Google Scholar 

  4. C. Borgs, J.T. Chayes, L. Lovász, V.T. Sós, and K. Vesztergombi, “Convergent sequences of dense graphs I: subgraph frequencies, metric properties and testing”, Advances in Mathematics 219 (6) (2008), 1801–1851.

    Google Scholar 

  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”, Annals of Mathematics 176 (2012), 151–219.

    Google Scholar 

  6. F. Chung and L. Lu, “The diameter of sparse random graphs”, Advances in Applied Mathematics 26 (2001), 257–279.

    Google Scholar 

  7. L. Devroye and N. Fraiman, “Connectivity threshold of inhomogeneous random graphs”, Random Structures & Algorithms 45 (3) (2014), 408–420.

    Google Scholar 

  8. Z. Füredi and J. Komlós, “The eigenvalues of random symmetric matrices”, Combinatorica 1 (3) (1981), 233–241.

    Google Scholar 

  9. V. Klee and D. Larman, “Diameters of random graphs”, Canadian Journal of Mathematics 33 (1981), 618–640.

    Google Scholar 

  10. L. Lovász, “Large networks and graph limits”, American Mathematical Society, 2012.

    Google Scholar 

  11. L. Lovász and B. Szegedy, “Limits of dense graph sequences”, Journal of Combinatorial Theory, Series B 96 (6) (2006), 933–957.

    Google Scholar 

  12. V. Marchenko and L. Pastur, “Distribution of eigenvalues for some sets of random matrices”, Math. USSR Sb. 1 (4) (1967), 457–483.

    Google Scholar 

  13. A. Nachmias and Y. Peres, “Critical random graphs: diameter and mixing time”, The Annals of Probability 36 (4) (2008), 1267–1286.

    Google Scholar 

  14. O. Riordan and N. Wormald, “The diameter of sparse random graphs”, Combinatorics, Probability & Computing 19 (5–6) (2010), 835–926.

    Google Scholar 

  15. B. Söderberg, “General formalism for inhomogeneous random graphs”, Physical review E 66 (6) (2002), 066121.

    Google Scholar 

  16. E. Wigner, “On the distribution of the roots of certain symmetric matrices”, Annals of Mathemathics 67 (1958), 325–328.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, 19104, Philadelphia, PA, USA

    Nicolas Fraiman

  2. Laboratoire J-A Dieudonné, Université Nice Sophia-Antipolis, Parc Valrose, 06108, Nice Cedex 02, France

    Dieter Mitsche

Authors
  1. Nicolas Fraiman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dieter Mitsche
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicolas Fraiman .

Editor information

Editors and Affiliations

  1. Departament de Ciències de la Computació, Universitat Politècnica de Catalunya, Barcelona, Spain

    Josep Díaz

  2. Department of Mathematics, National and Kapodistrian University, Zografos, Greece

    Lefteris Kirousis

  3. Department of Econometrics, University of Barcelona, Barcelona, Spain

    Luis Ortiz-Gracia

  4. Departament de Ciències de la Computació, Universitat Politècnica de Catalunya, Barcelona, Spain

    Maria Serna

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Fraiman, N., Mitsche, D. (2017). Metric and Spectral Properties of Dense Inhomogeneous Random Graphs. In: Díaz, J., Kirousis, L., Ortiz-Gracia, L., Serna, M. (eds) Extended Abstracts Summer 2015. Trends in Mathematics(), vol 6. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-51753-7_7

Download citation

Publish with us

Policies and ethics

Search

Navigation