Skip to main content
SpringerLink
Log in

Combinatorial Laplacian and entropy of simplicial complexes associated with complex networks

  • Regular Article
  • Published:
The European Physical Journal Special Topics Aims and scope Submit manuscript

Abstract

Simplicial complexes represent useful and accurate models of complex networks and complex systems in general. We explore the properties of spectra of combinatorial Laplacian operator of simplicial complexes and show its relationship with connectivity properties of the Q-vector and with connectivities of cliques in the simplicial clique complex. We demonstrate the need for higher order analysis in complex networks and compare the results with ordinary graph spectra. Methods and results are obtained using social network of the Zachary karate club.

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. R. Albert, A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002)

    Article  ADS  MATH  Google Scholar 

  2. S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D.-U. Hwang, Phys. Rep. 424, 175 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  3. S.N. Dorogovtsev, J.F.F. Mendes, Evolution of Networks: From Biological Nets to the Internet and WWW (Oxford University Press, Oxford, 2003)

  4. F. Harary, Graph Theory (Perseus, Cambridge, MA, 1995)

  5. R.H. Atkin, Mathematical structure in human affairs (Heinemann, London, 1974)

  6. R.W. Sharpe, Differential Geometry: Cartan’s genralization of Klein’s Erlangen Program (Springer-Verlag, New York, 1997)

  7. E. Gawlik, P. Mullen, D. Pavlov, J.E. Marsden, M. Desbrun, Physica D 240, 1724 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation (W.H. Freeman, San Francisco, 1973)

  9. W.W. Zachary, J. Anthropol. Res. 33, 452 (1977)

    Google Scholar 

  10. J.R. Munkres, Elements of Algebraic Topology (Addison-Wesley Publishing, California, 1984)

  11. S. Fortunato, Phys. Rep. 486, 75 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  12. G. Palla, I. Derényi, I. Farkas, T. Vicsek, Nature 435, 814 (2005)

    Article  ADS  Google Scholar 

  13. D. Kozlov, Combinatorial Algebraic Topology (Algorithms and Computation in Mathematics, Springer-Verlag, Berlin Heidelberg, 2008)

  14. S. Maletić, M. Rajković, D. Vasiljević, Lect. Notes Computer Sci., Springer 5102, 568 (2008)

    Article  Google Scholar 

  15. S. Maletić, D. Horak, M. Rajković, Lect. Notes Computer Sci., Springer 5102, 508 (2008)

    Google Scholar 

  16. D. Horak, S. Maletić, M. Rajković, J. Stat. Mech. 03, P03034 (2009)

    Article  Google Scholar 

  17. A.M. Duval, V. Reiner, Trans. Amer. Math. Soc. 354, 4313 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. T.E. Goldberg, Combinatorial Laplacians of Simplicial Complexes (Annandale-on-Hudson, New York, 2002)

  19. B. Mohar, Graph Theory, Combin., Appl. 2, 871 (1991)

    MathSciNet  Google Scholar 

  20. F.M. Atay, T. Biyikoğlu, Phys. Rev. E 72, 016217 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  21. A. Arenas, A. Díaz-Guilera, C.J. Pérez-Vicente, Phys. Rev. Lett. 96, 114102 (2006)

    Article  ADS  Google Scholar 

  22. M.E.J. Newman, Phys. Rev. E 74, 036104 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  23. A. Muhammad, M. Egerstedt, Proceedings of the 17th International Symposium on Mathematical Theory and Systems, Kyoto, Japan (2006), p. 1024

  24. D. Horak, J. Jost [arXiv: 1105.2712]

  25. D. Horak [arXiv: 1111.1836]

  26. G. Bianconi, Europhys. Lett. 81, 28005 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  27. G. Bianconi, A.C.C. Coolen, C.J. Perez Vicente, Phys. Rev. E 78, 016114 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  28. G. Bianconi, Phys. Rev. E 79, 036114 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  29. S.L. Braunstein, S. Ghosh, S. Severini, Ann. Combin. 10, 291 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. F.Passerini, S.Severini [arXiv:0812.2597v1]

  31. R.Atkin, Int. J. Man-Machine Stud. 4, 341 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  32. J.H.Johnson, Environ. Plann. B 8, 73 (1981)

    Article  Google Scholar 

  33. W.V.D.Hodge, The Theory and applications of harmonic integrals (Cambridge at the University Press, 1952)

  34. J.Friedman, Proc. 28th Annual ACM Symposium, Theory and Computations (1996), p. 386

  35. S.H.Friedberg, A.J.Insel, L.E.Spence, Linear Algebra (Prentice-Hall, Englewood Cliffs, NJ, 1997)

  36. F.R.K.Chung, Spectral Graph Theory, CBMS Regional Conference Series in Mathematics, No. 92 (American Mathematical Society, 1996)

  37. A.Banerjee, J.Jost, Discr. Appl. Math. 157, 2425 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. C.Bron, J.Kerbosch, Comm. ACM 16, 575 (1973)

    Article  MATH  Google Scholar 

  39. T.S.Evans, J. Stat. Mech. P12037 (2010)

Download references

Author information

Authors and Affiliations

  1. Institute of Nuclear Sciences Vinča, University of Belgrade, PO Box 522, 11001, Belgrade, Serbia

    S. Maletić & M. Rajković

Authors
  1. S. Maletić
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Rajković
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Maletić.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Maletić, S., Rajković, M. Combinatorial Laplacian and entropy of simplicial complexes associated with complex networks. Eur. Phys. J. Spec. Top. 212, 77–97 (2012). https://doi.org/10.1140/epjst/e2012-01655-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjst/e2012-01655-6

Keywords

Search

Navigation