Isoperimetric inequalities in simplicial complexes

Abstract

In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a complex, and the spectrum of the high dimensional Laplacian defined by Eckmann. In particular, we present a Cheeger-type inequality, and a high-dimensional Expander Mixing Lemma. As a corollary, using the work of Pach, we obtain a connection between spectral properties of complexes and Gromov’s notion of geometric overlap. Using the work of Gundert and Wagner, we give an estimate for the combinatorial expansion and geometric overlap of random Linial-Meshulam complexes.

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

References

  1. [1]

    R. Aharoni, E. Berger and R. Meshulam: Eigenvalues and homology of ag complexes and vector representations of graphs, Geometric and functional analysis 15 (2005), 555–566.

    MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    N. Alon and F. R. K. Chung: Explicit construction of linear sized tolerant networks, Discrete Mathematics 72 (1988), 15–19.

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    N. Alon: Eigenvalues and expanders, Combinatorica 6 (1986), 83–96.

    MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    N. Alon and V. D. Milman: λ1, isoperimetric inequalities for graphs, and superconcentrators, Journal of Combinatorial Theory, Ser. B 38 (1985), 73–88.

    MathSciNet  Article  MATH  Google Scholar 

  5. [5]

    M. Blum, R. M. Karp, O. Vorneberger, C. H. Papadimitriou and M. Yan-nakakis: Complexity of testing whether a graph is a superconcentrator, Information Processing Letters 13 (1981), 164–167.

    MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    Y. Bilu and N. Linial: Lifts, discrepancy and nearly optimal spectral gap, Combinatorica 26 (2006), 495–519.

    MathSciNet  Article  MATH  Google Scholar 

  7. [7]

    R. Beigel, G. Margulis and D. A. Spielman: Fault diagnosis in a small constant number of parallel testing rounds, in: Proceedings of the fifth annual ACM symposium on Parallel algorithms and architectures, ACM, 1993, 21–29.

    Google Scholar 

  8. [8]

    P. Buser: A note on the isoperimetric constant, Ann. Sci. École Norm. Sup. 15 (1982), 213–230.

    MathSciNet  MATH  Google Scholar 

  9. [9]

    J. Cheeger: A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis 195 (1970), 199.

    MathSciNet  MATH  Google Scholar 

  10. [10]

    F. Chung: The Laplacian of a hypergraph, Expanding Graphs (Joel Friedman, ed.), DIMACS, 10, AMS, 1993, 21–36.

    Google Scholar 

  11. [11]

    F. R. K. Chung: Spectral graph theory, CBMS, Amer Mathematical Society, 1997.

    MATH  Google Scholar 

  12. [12]

    D. Dotterrer and M. Kahle: Coboundary expanders, Journal of Topology and Analysis 4 (2012), 499–514.

    MathSciNet  Article  MATH  Google Scholar 

  13. [13]

    A. Duval, C. Klivans and J. Martin: Simplicial matrix-tree theorems, Transactions of the American Mathematical Society 361 (2009), 6073–6114.

    MathSciNet  Article  MATH  Google Scholar 

  14. [14]

    J. Dodziuk: Finite-difference approach to the Hodge theory of harmonic forms, American Journal of Mathematics 98 (1976), 79–104.

    MathSciNet  Article  MATH  Google Scholar 

  15. [15]

    J. Dodziuk: Difference equations, isoperimetric inequality and transience of certain random walks, Trans. Amer. Math. Soc. 284 (1984).

  16. [16]

    B. Eckmann: Harmonische funktionen und randwertaufgaben in einem komplex, Commentarii Mathematici Helvetici 17 (1944), 240–255.

    MathSciNet  Article  Google Scholar 

  17. [17]

    P. Erdős and A. Rényi: On random graphs, Publicationes Mathematicae Debrecen 6 (1959), 290–297.

    MathSciNet  MATH  Google Scholar 

  18. [18]

    P. Erdős and A. Rényi: On the evolution of random graphs, Bull. Inst. Internat. Statist 38 (1961), 343–347.

    MathSciNet  MATH  Google Scholar 

  19. [19]

    J. Fox, M. Gromov, V. Lafforgue, A. Naor and J. Pach: Overlap properties of geometric expanders, J. Reine Angew. Math. 671 (2012), 49–83.

    MathSciNet  MATH  Google Scholar 

  20. [20]

    J. Friedman and N. Pippenger: Expanding graphs contain all small trees, Combinatorica 7 (1987), 71–76.

    MathSciNet  Article  MATH  Google Scholar 

  21. [21]

    J. Friedman: Computing Betti numbers via combinatorial Laplacians, Algorithmica 21 (1998), 331–346.

    MathSciNet  Article  MATH  Google Scholar 

  22. [22]

    J. Friedman: A proof of Alon's second eigenvalue conjecture and related problems, Mem. Amer. Math. Soc. 195 (2008).

  23. [23]

    J. Friedman and A. Wigderson: On the second eigenvalue of hypergraphs, Combinatorica 15 (1995), 43–65.

    MathSciNet  Article  MATH  Google Scholar 

  24. [24]

    H. Garland: p-adic curvature and the cohomology of discrete subgroups of p-adic groups, The Annals of Mathematics 97 (1973), 375–423.

    MathSciNet  Article  MATH  Google Scholar 

  25. [25]

    M. Gromov: Singularities, expanders and topology of maps. Part 2: from combinatorics to topology via algebraic isoperimetry, Geometric And Functional Analysis 20 (2010), 416–526.

  26. [26]

    A. Gundert and M. Szedlák: Higher dimensional discrete Cheeger inequalities, Annual Symposium on Computational Geometry (New York, NY, USA), SOCG14, ACM, 2014.

    Google Scholar 

  27. [27]

    A. Gundert and U. Wagner: On Laplacians of random complexes, in: Proceedings of the 2012 symposuim on Computational Geometry, ACM, 2012, 151–160.

    Google Scholar 

  28. [28]

    S. Hoory, N. Linial and A. Wigderson: Expander graphs and their applications, Bulletin of the American Mathematical Society 43 (2006), 439–562.

    MathSciNet  Article  MATH  Google Scholar 

  29. [29]

    S. Janson: On concentration of probability, Contemporary combinatorics 10 (2002), 1–9.

    MathSciNet  MATH  Google Scholar 

  30. [30]

    W. Kook, V. Reiner and D. Stanton: Combinatorial Laplacians of matroid complexes, Journal of the American Mathematical Society 13 (2000), 129–148.

    MathSciNet  Article  MATH  Google Scholar 

  31. [31]

    N. Linial and R. Meshulam: Homological connectivity of random 2-complexes, Combinatorica 26 (2006), 475–487.

    MathSciNet  Article  MATH  Google Scholar 

  32. [32]

    A. Lubotzky, R. Phillips and P. Sarnak: Ramanujan graphs, Combinatorica 8 (1988), 261–277.

    MathSciNet  Article  MATH  Google Scholar 

  33. [33]

    A. Lubotzky, B. Samuels and U. Vishne: Ramanujan complexes of type Ãd, Israel Journal of Mathematics 149 (2005), 267–299.

    MathSciNet  Article  MATH  Google Scholar 

  34. [34]

    A. Lubotzky: Discrete groups, expanding graphs and invariant measures, vol. 125, Birkhauser, 2010.

  35. [35]

    A. Lubotzky: Expander graphs in pure and applied mathematics, Bull. Amer. Math. Soc. 49 (2012), 113–162.

    MathSciNet  Article  MATH  Google Scholar 

  36. [36]

    D. W. Matula and F. Shahrokhi: Sparsest cuts and bottlenecks in graphs, Discrete Applied Mathematics 27 (1990), 113–123.

    MathSciNet  Article  MATH  Google Scholar 

  37. [37]

    A. Marcus, D. A. Spielman and N. Srivastava: Interlacing families I: Bipartite Ramanujan graphs of all degrees, arXiv preprint arXiv:1304.4132, (2013).

    Google Scholar 

  38. [38]

    R. Meshulam and N. Wallach: Homological connectivity of random k-dimensional complexes, Random Structures & Algorithms 34 (2009), 408–417.

    MathSciNet  Article  MATH  Google Scholar 

  39. [39]

    J. Matoušsek and U. Wagner: On Gromov's method of selecting heavily covered points, Discrete & Computational Geometry 52 (2014), 1–33.

    MathSciNet  Article  MATH  Google Scholar 

  40. [40]

    A. Nilli: On the second eigenvalue of a graph, Discrete Mathematics 91 (1991), 207–210.

    MathSciNet  Article  MATH  Google Scholar 

  41. [41]

    I. Newman and Y. Rabinovich: On multiplicative λ-approximations and some geometric applications, in: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '12, SIAM, 2012, 51–67.

    Google Scholar 

  42. [42]

    R. I. Oliveira: Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges, Arxiv preprint ArXiv:0911.0600v2 (2010).

    Google Scholar 

  43. [43]

    J. Pach: A Tverberg-type result on multicolored simplices, Computational Geometry 10 (1998), 71–76.

    MathSciNet  Article  MATH  Google Scholar 

  44. [44]

    O. Parzanchevski and R. Rosenthal: Simplicial complexes: spectrum, homology and random walks, arXiv preprint arXiv:1211.6775 (2012).

    Google Scholar 

  45. [45]

    D. Puder: Expansion of random graphs: New proofs, new results, Inventiones mathematicae (2014), 1–64.

    Google Scholar 

  46. [46]

    J. Steenbergen, C. Klivans and S. Mukherjee: A Cheeger-type inequality on simplicial complexes, Advances in Applied Mathematics 56 (2014), 56–77.

    MathSciNet  Article  MATH  Google Scholar 

  47. [47]

    R. M. Tanner: Explicit concentrators from generalized n-gons, SIAM Journal on Algebraic and Discrete Methods 5 (1984), 287.

    MathSciNet  Article  MATH  Google Scholar 

  48. [48]

    T. Tao: Basic theory of expander graphs, http://terrytao.wordpress.com/2011/12/02/245b-notes-1-basic-theory-of-expander-graphs/, 2011.

    Google Scholar 

  49. [49]

    A. Żuk: La propriété (T) de Kazhdan pour les groupes agissant sur les polyedres, in: Comptes rendus de l'Académie des sciences, Série 1, Mathématique 323 (1996), 453–458.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ori Parzanchevski.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Parzanchevski, O., Rosenthal, R. & Tessler, R.J. Isoperimetric inequalities in simplicial complexes. Combinatorica 36, 195–227 (2016). https://doi.org/10.1007/s00493-014-3002-x

Download citation

Keywords

  • Random Graph
  • Simplicial Complex
  • Isoperimetric Inequality
  • Expander Graph
  • Complete Skeleton