Skip to main content
SpringerLink
Log in

Basics of Graph Limit Theory

  • Chapter
  • First Online:
Large Deviations for Random Graphs

Part of the book series: Lecture Notes in Mathematics ((LNMECOLE,volume 2197))

Abstract

This chapter summarizes some basic results from graph limit theory. The only background assumed here is the list of results from the previous chapter.

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 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
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. Aldous, D. (1981). Representations for partially exchangeable arrays of random variables. Journal of Multivariate Analysis, 11(4), 581–598.

    Article  MathSciNet  MATH  Google Scholar 

  2. Austin, T. (2008). On exchangeable random variables and the statistics of large graphs and hypergraphs. Probability Surveys, 5, 80–145.

    Article  MathSciNet  MATH  Google Scholar 

  3. Bollobás, B., & Riordan, O. (2009). Metrics for sparse graphs. In Surveys in combinatorics 2009, vol. 365, London mathematical society of lecture note series (pp. 211–287). Cambridge: Cambridge University Press.

    Google Scholar 

  4. Borgs, C., Chayes, J., Lovász, L., Sós, V. T., & Vesztergombi, K. (2006). Counting graph homomorphisms. In Topics in discrete mathematics, vol. 26, Algorithms and combinatorics (pp. 315–371). Berlin: Springer.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Borgs, C., Chayes, J., Lovász, L., Sós, V. T., & Vesztergombi, K. (2012). Convergent sequences of dense graphs. II. Multiway cuts and statistical physics. Annals of Mathematics, 176(1), 151–219.

    Article  MathSciNet  MATH  Google Scholar 

  7. Borgs, C., Chayes, J. T., Cohn, H., & Zhao, Y. (2014). An L p theory of sparse graph convergence I: Limits, sparse random graph models, and power law distributions. arXiv preprint arXiv:1401.2906

    Google Scholar 

  8. Borgs, C., Chayes, J. T., Cohn, H., & Zhao, Y. (2014). An L p theory of sparse graph convergence II: LD convergence, quotients, and right convergence. arXiv preprint arXiv:1408.0744

    Google Scholar 

  9. Diaconis, P., & Janson, S. (2008). Graph limits and exchangeable random graphs. Rendiconti di Matematica e delle sue Applicazioni. Serie VII, 28(1), 33–61. http://www1.mat.uniroma1.it/ricerca/rendiconti/

    MathSciNet  MATH  Google Scholar 

  10. Freedman, M., Lovász, L., & Schrijver, A. (2007). Reflection positivity, rank connectivity, and homomorphism of graphs. Journal of the American Mathematical Society, 20, 37–51 (electronic).

    Google Scholar 

  11. Frieze, A., & Kannan, R. (1999). Quick approximation to matrices and applications. Combinatorica, 19, 175–220.

    Article  MathSciNet  MATH  Google Scholar 

  12. Hoover, D. N. (1982). Row-column exchangeability and a generalized model for probability. Exchangeability in probability and statistics (Rome, 1981), (pp. 281–291). Amsterdam: North-Holland.

    Google Scholar 

  13. Kallenberg, O. (2005). Probabilistic symmetries and invariance principles. New York: Springer.

    MATH  Google Scholar 

  14. Lovász, L. (2006). The rank of connection matrices and the dimension of graph algebras. European Journal of Combinatorics, 27, 962–970.

    Article  MathSciNet  MATH  Google Scholar 

  15. Lovász, L. (2007). Connection matrices. In Combinatorics, complexity, and chance, vol. 34, Oxford lecture series in mathematics and its applications (pp. 179–190). Oxford: Oxford University Press.

    Google Scholar 

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

    Book  MATH  Google Scholar 

  17. Lovász, L., & Sós, V. T. (2008). Generalized quasirandom graphs. Journal of Combinatorial Theory, Series B, 98, 146–163.

    Article  MathSciNet  MATH  Google Scholar 

  18. Lovász, L., & Szegedy, B. (2006). Limits of dense graph sequences. Journal of Combinatorial Theory, Series B, 96(6), 933–957.

    Article  MathSciNet  MATH  Google Scholar 

  19. Lovász, L., & Szegedy, B. (2007). Szemerédi’s lemma for the analyst. Geometric and Functional Analysis, 17, 252–270.

    Article  MathSciNet  MATH  Google Scholar 

  20. Lovász, L., & Szegedy, B. (2010). Testing properties of graphs and functions. Israel Journal of Mathematics, 178, 113–156.

    Article  MathSciNet  MATH  Google Scholar 

  21. Lovász, L., & Szegedy, B. (2009). Contractors and connectors of graph algebras. Journal of Graph Theory, 60 11–30.

    Article  MathSciNet  MATH  Google Scholar 

  22. Szegedy, B. (2011). Limits of kernel operators and the spectral regularity lemma. European Journal of Combinatorics, 32(7), 1156–1167.

    Article  MathSciNet  MATH  Google Scholar 

  23. Szemerédi, E. (1978). Regular partitions of graphs. In Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), pp. 399–401, Colloq. Internat. CNRS, 260, CNRS, Paris.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, Stanford University, Stanford, CA, USA

    Sourav Chatterjee

Authors
  1. Sourav Chatterjee
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Chatterjee, S. (2017). Basics of Graph Limit Theory. In: Large Deviations for Random Graphs. Lecture Notes in Mathematics(), vol 2197. Springer, Cham. https://doi.org/10.1007/978-3-319-65816-2_3

Download citation

Publish with us

Policies and ethics

Search

Navigation