Advertisement

On the Robustness of Complex Networks by Using the Algebraic Connectivity

  • A. Jamakovic
  • Piet Van Mieghem
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4982)

Abstract

The second smallest eigenvalue of the Laplacian matrix, also known as the algebraic connectivity, plays a special role for the robustness of networks since it measures the extent to which it is difficult to cut the network into independent components. In this paper we study the behavior of the algebraic connectivity in a well-known complex network model, the Erdős-Rényi random graph. We estimate analytically the mean and the variance of the algebraic connectivity by approximating it with the minimum nodal degree. The resulting estimate improves a known expression for the asymptotic behavior of the algebraic connectivity [18].Simulations emphasize the accuracy of the analytical estimation, also for small graph sizes. Furthermore, we study the algebraic connectivity in relation to the graph’s robustness to node and link failures, i.e. the number of nodes and links that have to be removed in order to disconnect a graph. These two measures are called the node and the link connectivity. Extensive simulations show that the node and the link connectivity converge to a distribution identical to that of the minimal nodal degree, already at small graph sizes. This makes the minimal nodal degree a valuable estimate of the number of nodes or links whose deletion results into disconnected random graph. Moreover, the algebraic connectivity increases with the increasing node and link connectivity, justifies the correctness of our definition that the algebraic connectivity is a measure of the robustness in complex networks.

Keywords

Complex Network Random Graph Link Failure Laplacian Matrix Graph Size 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover Publications, Inc, NY (1999)zbMATHGoogle Scholar
  2. 2.
    Albert, R., Barabasi, A.-L.: Statistical mechanics of complex networks. Reviews of Modern Physics 74(1) (January 2002)Google Scholar
  3. 3.
    Bai, Z., Demmel, J., Dongarra, J., Ruhe, A., van der Vorst, H.: Templates for the solution of Algebraic Eigenvalue Problems: A Practical Guide. SIAM, Philadelphia (2000)Google Scholar
  4. 4.
    Barabasi, A.-L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barabasi, A.-L.: Linked. The new science of networks. Perseus, Cambridge (2002)Google Scholar
  6. 6.
    Bollobás, B.: Random graphs, 2nd edn. Cambridge University Press, Cambridge (2001)CrossRefzbMATHGoogle Scholar
  7. 7.
    Bollobás, B., Thomason, A.G.: Random graphs of small order. Random graphs 1983. Annals of Discrete Mathematics 28, 47–97 (1985)zbMATHGoogle Scholar
  8. 8.
    Castro, M., Costa, M., Rowstron, A.: Should We Build Gnutella on a Structured Overlay. ACM SIGCOMM Computer Communications Review 34(1), 131–136 (2004)CrossRefGoogle Scholar
  9. 9.
    Chung, F.R.K.: Spectral Graph Theory. In: Conference Board of the Mathematical Sciences, American Mathematical Society, Providence, RI, vol. 92 (1997)Google Scholar
  10. 10.
    Cvetkovic, D.M., Doob, M., Sachs, H.: Spectra of Graphs, Theory and Applications, 3rd edn. Johann Ambrosius Barth, Heidelberg (1995)zbMATHGoogle Scholar
  11. 11.
    Dorogovtsev, S.N., Mendes, J.F.F.: Evolution of Networks. In: From Biological Nets to the Internet and WWW, Oxford University Press, Oxford (2003)CrossRefGoogle Scholar
  12. 12.
    Erdős, P., Rényi, A.: On random graphs. Publicationes Mathematicae 6, 290–297 (1959)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Fiedler, M.: Algebraic connectivity of graphs. Czechoslovak Mathematical Journal 23, 298–305 (1973)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Fuhrmann, T.: On the Topology of Overlay Networks. In: Proceedings of 11th IEEE International Conference on Networks (ICON), pp. 271–276 (2003)Google Scholar
  15. 15.
    Gibbons, A.: Algorithmic Graph Theory. Cambridge University Press, Cambridge (1985)zbMATHGoogle Scholar
  16. 16.
    Jamakovic, A., Van Mieghem, P.: On the robustness of complex networks by using the algebraic connectivity, Technical Report (2007)Google Scholar
  17. 17.
    Janson, S., Knuth, D.E., Luczak, T., Pittel, B.: The birth of the giant component. Random Structures & Algorithms 4, 233–358 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Juhāz, F.: The asymptotic behaviour of Fiedler’s algebraic connectivity for random graphs. Discrete Mathematics 96, 59–63 (1991)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Merris, R.: Laplacian matrices of graphs: a survey. In: Linear Algebra Applications, vol. 197,198, pp. 143–176 (1994)Google Scholar
  20. 20.
    Merris, R.: A survey of graph Laplacians. In: Linear and Multilinear Algebra, vol. 39, pp. 19–31 (1995)Google Scholar
  21. 21.
    Mohar, B., Alavi, Y., Chartrand, G., Oellermann, O.R., Schwenk, A.J.: The Laplacian spectrum of graphs. Graph Theory, Combinatorics and Applications 2, 871–898 (1991)MathSciNetGoogle Scholar
  22. 22.
    Mohar, B.: Laplace eigenvalues of graphs: a survey. Discrete Mathematics 109, 198, 171–183 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mohar, B., Hahn, G., Sabidussi, G.: Some applications of Laplace eigenvalues of graphs. Graph Symmetry: Algebraic Methods and Applications, NATO ASI Ser. C 497, 225–275 (1997)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Newman, M.: The structure and function of complex networks. SIAM Review 45, 167–256 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Skiena, S.: Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Addison-Wesley, Reading (1990)zbMATHGoogle Scholar
  26. 26.
    Van Mieghem, P.: Performance Analysis of Computer Systems and Networks. Cambridge University Press, Cambridge (2006)CrossRefzbMATHGoogle Scholar
  27. 27.
    Watts, D.J., Strogatz, S.H.: Collective dynamics of small-world networks. Nature 393, 440–442 (1999)CrossRefzbMATHGoogle Scholar
  28. 28.
    Watts, D.J.: Small Worlds. The Dynamics of Networks Between Order and Randomness. Princeton University Press, Princeton (1999)zbMATHGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2008

Authors and Affiliations

  • A. Jamakovic
    • 1
  • Piet Van Mieghem
    • 1
  1. 1.Electrical Engineering, Mathematics and Computer ScienceDelft University of TechnologyDelftThe Netherlands

Personalised recommendations