The intrinsic dimensionality of graphs

Abstract

We resolve the following conjecture raised by Levin together with Linial, London, and Rabinovich [Combinatorica, 1995]. For a graph G, let dim(G) be the smallest d such that G occurs as a (not necessarily induced) subgraph of ℤ d , the infinite graph with vertex set ℤd and an edge (u, v) whenever ∥uv = 1. The growth rate of G, denoted ρ G , is the minimum ρ such that every ball of radius r > 1 in G contains at most r ρ vertices. By simple volume arguments, dim(G) = Ω(ρ G ). Levin conjectured that this lower bound is tight, i.e., that dim(G) = O(ρ G ) for every graph G.

Previously, it was unknown whether dim(G) could be bounded above by any function of ρ G . We show that a weaker form of Levin’s conjecture holds by proving that dim(G) = O(ρ G log ρ G ) for any graph G. We disprove, however, the specific bound of the conjecture and show that our upper bound is tight by exhibiting graphs for which dim(G) = Ω(ρ G log ρ G ). For several special families of graphs (e.g., planar graphs), we salvage the strong form, showing that dim(G) = O(ρ G ). Our results extend to a variant of the conjecture for finite-dimensional Euclidean spaces posed by Linial and independently by Benjamini and Schramm.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    N. Alon and J. H. Spencer: The probabilistic method, Wiley-Interscience [John Wiley & Sons], New York, second edition, 2000.

    Google Scholar 

  2. [2]

    P. Assouad: Plongements lipschitziens dans Rn, Bull. Soc. Math. France111(4) (1983), 429–448.

    MathSciNet  MATH  Google Scholar 

  3. [3]

    Y. Bartal: Probabilistic approximation of metric spaces and its algorithmic applications, in 37th Annual Symposium on Foundations of Computer Science, pages 184–193, IEEE, 1996.

  4. [4]

    J. A. Bondy and U. S. R. Murty: Graph theory with applications, American Elsevier Publishing Co., Inc., New York, 1976.

    Google Scholar 

  5. [5]

    A. Brandstädt, V. Chepoi and F. Dragan: Distance approximating trees for chordal and dually chordal graphs, J. Algorithms30(1) (1999), 166–184.

    Article  MathSciNet  MATH  Google Scholar 

  6. [6]

    V. Chepoi and F. Dragan: A note on distance approximating trees in graphs, European J. Combin.21(6) (2000), 761–766.

    Article  MathSciNet  MATH  Google Scholar 

  7. [7]

    F. R. K. Chung: Labelings of graphs, in Selected topics in graph theory, 3, pages 151–168, Academic Press, San Diego, CA, 1988.

    Google Scholar 

  8. [8]

    P. Erdős, F. Harary and W. T. Tutte: On the dimension of a graph, Mathematika12 (1965), 118–122.

    MathSciNet  Article  Google Scholar 

  9. [9]

    U. Feige: Approximating the bandwidth via volume respecting embeddings, J. Comput. System Sci.60(3) (2000), 510–539.

    Article  MathSciNet  MATH  Google Scholar 

  10. [10]

    J. Fakcharoenphol and K. Talwar: An improved decomposition theorem for graphs excluding a fixed minor, in Proceedings of 6th Workshop on Approximation, Randomization, and Combinatorial Optimization, Springer Lecture Notes in Computer Science 2764, 36–46, 2003.

  11. [11]

    A. Gupta, R. Krauthgamer and J. R. Lee: Bounded geometries, fractals, and low-distortion embeddings; in Proceedings of the 44th Annual Symposium on Foundations of Computer Science, 2003.

  12. [12]

    J. Heinonen: Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001.

    Google Scholar 

  13. [13]

    P. Indyk: Algorithmic applications of low-distortion geometric embeddings, in Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science, pages 10–33, October 2001.

  14. [14]

    P. Klein, S. A. Plotkin and S. Rao: Excluded minors, network decomposition, and multicommodity flow; in 25th Annual ACM Symposium on Theory of Computing, pages 682–690, May 1993.

  15. [15]

    N. Linial: Variation on a theme of Levin, in Open Problems, Workshop on Discrete Metric Spaces and their Algorithmic Applications (J. Matoušek, ed.), Haifa, March 2002.

  16. [16]

    N. Linial, E. London and Y. Rabinovich: The geometry of graphs and some of its algorithmic applications, Combinatorica15(2) (1995), 215–245.

    Article  MathSciNet  MATH  Google Scholar 

  17. [17]

    N. Linial and M. Saks: Low diameter graph decompositions, Combinatorica13(4) (1993), 441–454.

    Article  MathSciNet  MATH  Google Scholar 

  18. [18]

    L. Lovász and K. Vesztergombi: Geometric representations of graphs, in Paul Erdős, Proc. Conf., Budapest, 1999.

  19. [19]

    J. Matoušek: Lectures on discrete geometry, Vol. 212 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2002.

    Google Scholar 

  20. [20]

    R. Motwani and P. Raghavan: Randomized Algorithms, Cambridge University Press, 1995.

  21. [21]

    S. Rao: Small distortion and volume preserving embeddings for planar and Euclidean metrics, in Proceedings of the 15th Annual Symposium on Computational Geometry, pages 300–306, ACM, 1999.

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Robert Krauthgamer.

Additional information

Supported by NSF grant CCR-0121555 and by an NSF Graduate Research Fellowship.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Krauthgamer, R., Lee, J.R. The intrinsic dimensionality of graphs. Combinatorica 27, 551–585 (2007). https://doi.org/10.1007/s00493-007-2183-y

Download citation

Mathematics Subject Classification (2000)

  • 05C62
  • 51F99