Skip to main content
SpringerLink
Log in

Compact graphings

  • Published:
Acta Mathematica Hungarica Aims and scope Submit manuscript

Abstract

Graphings are special bounded-degree graphs on probability spaces, representing limits of graph sequences that are convergent in a local or local-global sense. A graphing is compact, if the underlying space is a compact metric space, the edge set is closed and nearby points have nearby graph neighborhoods. We prove some simple properties of compact graphings. Our main result is a procedure for embedding an arbitrary graphing into an essentially equivalent compact graphing.

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

Similar content being viewed by others

References

  1. D. Aldous and R. Lyons, Processes on unimodular random networks, Electron. J. Probab., 12, Paper 54 (2007), 1454–1508

  2. Baccelli, F., Haji-Mirsadeghi, M., Khezeli, A.: Eternal family trees and dynamics on unimodular random graphs. Contemp. Math. 719, 85–127 (2018)

    Article  MathSciNet  Google Scholar 

  3. I. Benjamini and O. Schramm, Recurrence of distributional limits of finite planar graphs, Electron. J. Probab., 6 (2001), paper No. 23, 13 pp

  4. G. Elek, Uniformly recurrent subgroups and simple C*-algebras, https://arxiv.org/abs/1704.02595

  5. G. Elek, Qualitative graph limit theory. Cantor dynamical systems and constant-time distributed algorithms, https://arxiv.org/abs/1812.07511

  6. G. Elek, Free minimal actions of countable groups with invariant probability measures, https://arxiv.org/abs/1805.11149

  7. Hatami, H., Lovász, L., Szegedy, B.: Limits of locally-globally convergent graph sequences. Geom. Func. Anal. 24, 269–296 (2014)

    Article  MathSciNet  Google Scholar 

  8. Kechris, A., Solecki, S., Todorčević, S.: Borel chromatic numbers. Adv. in Math. 141, 1–44 (1999)

    Article  MathSciNet  Google Scholar 

  9. L. Lovász, Large Networks and Graph Limits, Amer. Math. Soc. (Providence, R.I., 2012)

  10. Lovász, L., Szegedy, B.: Szemerédi's Lemma for the analyst. Geom. Func. Anal. 17, 252–270 (2007)

    Article  Google Scholar 

  11. L. Lovász and B. Szegedy, Regularity partitions and the topology of graphons, in: An Irregular Mind, Szemerédi is 70, J. Bolyai Math. Soc. and Springer-Verlag (2010), pp. 415–446

  12. Lovász, L., Szegedy, B.: The automorphism group of a graphon. J. Algebra 421, 136–166 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

I am grateful to the referee of the paper for pointing out a gap in the main proof, and to Francois Baccelli for calling my attention to the connection with their “No Finite Inclusion Lemma” in [2].

Author information

Authors and Affiliations

  1. Hungarian Academy of Sciences and Eötvös Loránd University, Budapest, Hungary

    L. Lovász

Authors
  1. L. Lovász
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. Lovász.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lovász, L. Compact graphings. Acta Math. Hungar. 161, 185–196 (2020). https://doi.org/10.1007/s10474-019-01010-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10474-019-01010-8

Key words and phrases

Mathematics Subject Classification

Search

Navigation