Advertisement

Classes of the Shortest Pathway Structures in Scale Free Networks

  • Kwang-Il Goh
  • Eulsik Oh
  • Chul-Min Ghim
  • Byungnam Kahng
  • Doochul Kim
Part I Network Structure
Part of the Lecture Notes in Physics book series (LNP, volume 650)

Abstract

We study a problem of data packet transport between a pair of vertices on scale-free network, and introduce load of a vertex as the accumulated sum of a fraction of data packets traveling along the shortest pathways between every pair of vertices. It is found that the load distributions for many real-world networks follow a power law with an exponent δ which is close to either 2.2(1) (class I) or 2.0 (class II), insensitive to different values of the degree exponent γ in the range, 2 < γ < 3. The classification of scale-free networks into the two classes may stem from the characteristics of the shortest pathways structures. While the shortest pathways between a pair of vertices are multiply connected in the class I, they are almost singly connected in the class II. Such distinct topological features of the shortest pathways produce different behaviors in diverse problems such as the distribution of diameter change by the removal of a single vertex. Finally, we remark that since the two numerical values of the load exponent are too close, it still remains an open question if they are really robust. Analytic solution resolving this controversial issue is needed.

Keywords

Load Distribution Betweenness Centrality Scale Free Network Protein Interaction Network Real World Network 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1. K. Ziemelis and L. Allen, Nature 410, 241 (2001) and following review articles on complex systems.Google Scholar
  2. 2. R. Gallagher and T. Appenzeller, Science 284, 87 (1999) and following viewpoint articles on complex systems.Google Scholar
  3. 3. S. H. Strogatz, Nature 410, 268 (2001).Google Scholar
  4. 4. R. Albert and A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002).Google Scholar
  5. 5. S. N. Dorogovtsev and J. F. F. Mendes, Evolution of Networks: From Biological Nets to the Internet and WWW (Oxford University Press, Oxford, 2003).Google Scholar
  6. 6. M. E. J. Newman, SIAM Review 45, 167 (2003).Google Scholar
  7. 7. P. Erdős and A. Rényi, Publ. Math. Inst. Hung. Acad.  Sci. Ser. A 5, 17 (1960).Google Scholar
  8. 8. A.-L. Barabási, R. Albert, and H. Jeong, Physica A 272, 173 (1999).Google Scholar
  9. 9. R. Albert, H. Jeong, and A.-L. Barabási, Nature 401, 130 (1999).Google Scholar
  10. 10. B. A. Huberman and L. A. Adamic, Nature 401, 131 (1999).Google Scholar
  11. 11. A. Broder, et al., Computer Networks 33, 309 (2000).Google Scholar
  12. 12. M. Faloutsos, P. Faloutsos, and C. Faloutsos, Comput. Commun. Rev. 29, 251 (1999).Google Scholar
  13. 13. R. Pastor-Satorras, A. Vázquez, and A. Vespignani, Phys. Rev. Lett. 87, 258701 (2001).Google Scholar
  14. 14. K.-I. Goh, B. Kahng, and D. Kim, Phys. Rev. Lett. 88, 108701 (2002).Google Scholar
  15. 15. S. Redner, Eur. Phys. J. B 4, 131 (1998).Google Scholar
  16. 16. M. E. J. Newman, Proc. Natl. Acad. Sci. USA 98, 404 (2001).Google Scholar
  17. 17. A.-L. Barabási, H. Jeong, R. Ravasz, Z. Neda, T. Vicsek, and A. Schubert, Physica A 311, 590-614 (2002).Google Scholar
  18. 18. H. Jeong, B. Tombor, R. Albert, Z. N. Oltvani, and A.-L. Barabási, Nature 407, 651 (2000).Google Scholar
  19. 19. A.-L. Barabási and R. Albert, Science 286, 509 (1999).Google Scholar
  20. 20. P. L. Krapivsky, S. Redner, and F. Leyvraz, Phys. Rev. Lett. 85, 4629 (2000).Google Scholar
  21. 21. S. N. Dorogovtsev, J. F. F. Mendes, A. N. Samukhin, Phys. Rev. Lett. 85, 4633 (2000).Google Scholar
  22. 22. K.-I. Goh, B. Kahng, and D. Kim, Phys. Rev. Lett. 87, 278701 (2001).Google Scholar
  23. 23. L. C. Freeman, Sociometry 40, 35 (1977).Google Scholar
  24. 24. M. E. J. Newman, Phys. Rev. E 64, 016132 (2001).Google Scholar
  25. 25. U. Brandes, J. Math. Sociol. 25, 163 (2001).Google Scholar
  26. 26. H. Jeong, S. P. Mason, A.-L. Barabási, and Z. N. Oltvai, Nature 411, 41 (2001).Google Scholar
  27. 27. T. Ito, T. Chiba, R. Ozawa, M. Yoshida, M. Hattori, and Y. Sakaki, Proc. Natl. Acad. Sci. USA 98, 4569 (2000).Google Scholar
  28. 28. B. A. Huberman and L. A. Adamic, e-print (cond-mat/9901071) (1999).Google Scholar
  29. 29. R. Kumar, P. Raghavan, S. Rajagopalan, D. Sivakumar, A. Tomkins, and E. Upfal, in Proc. IEEE FOCS 2000 (IEEE Computer Society Press, Los Alamitos, CA, 2000), pp. 57-65.Google Scholar
  30. 30. S. N. Dorogovtsev and J. F. F. Mendes, Phys. Rev. E 63, 025101(R) (2001).Google Scholar
  31. 31. R. Solé, R. Pastor-Satorras, E. Smith, and T. Kepler, Adv. Complex. Syst. 5, 43 (2002).Google Scholar
  32. 32. J. Kim, P. L. Krapivsky, B. Kahng, and S. Redner, Phys. Rev. E 66, 055101(R) (2002).Google Scholar
  33. 33. Meyer, D. (2001) University of Oregon Route Views Archive Project (http://archive.routeviews.org).Google Scholar
  34. 34. S. Jung, S. Kim, and B. Kahng, Phys. Rev. E 65, 056101 (2002).Google Scholar
  35. 35. K.-I. Goh, E. Oh, H. Jeong, B. Kahng and D. Kim, Proc. Natl. Acad. Sci. USA 99, 12583 (2002).Google Scholar
  36. 36. G. Szabó, M. Alava, and J. Kertész, Phys. Rev. E 66, 026101 (2002).Google Scholar
  37. 37. K.-I. Goh, C.-M. Ghim, B. Kahng, and D. Kim, Phys. Rev. Lett. 91, 189804 (2003).Google Scholar
  38. 38. M. Barthélemy, Phys. Rev. Lett. 91, 189803 (2003).Google Scholar
  39. 39. M. E. J. Newman, Phys. Rev. Lett. 89, 208701 (2002).Google Scholar
  40. 40. M. E. J. Newman, Phys. Rev. E 67, 026126 (2003).Google Scholar
  41. 41. P. L. Krapivsky and S. Redner, Phys. Rev. E 63, 066123 (2001).Google Scholar
  42. 42. S. N. Dorogovtsev, J. F. F. Mendes, and A. N. Samukhin, e-print (cond-mat/0206467) (2002).Google Scholar
  43. 43. http://www.imdb.com.Google Scholar
  44. 44. M. Molloy and B. Reed, Random Structures and Algorithms 6, 161 (1995); Z. Burda and A. Krzywicki, Phys. Rev. E 67, 046118 (2003).Google Scholar
  45. 45. Given a degree sequence ${ cal D} equiv {ki} (i=1,..., N)$, we generate a maximally random network whose degree sequence is $ cal D$, with a constraint that any self-loops and multiple edges are forbidden. By choosing $cal D ={ lfloor (N/i) 1/( gamma-1) rfloor }, we get a SF network with degree exponent gamma, where $ lfloor cdot rfloor$ is the floor function.Google Scholar
  46. 46. G. Parisi, Physica A 263, 557 (1999).Google Scholar
  47. 47. J. P. Sethna, K. A. Dahmen, and C. R. Myers, Nature 410, 242 (2001) and references therein.Google Scholar
  48. 48. R. Albert, H. Jeong, and A.-L. Barabási, Nature 406, 378 (2000).Google Scholar
  49. 49. Z. Liu, Y.-C. Lai, and N. Ye, Phys. Rev. E 66, 036112 (2002).Google Scholar
  50. 50. G. Bianconi and A.-L. Barabási, Europhys. Lett. 54, 436 (2001).Google Scholar
  51. 51. K.-I. Goh, B. Kahng, and D. Kim, e-print (q.bio-MN/0312009).Google Scholar
  52. 52. R. Cohen and S. Havlin, Phys. Rev. Lett. 90, 058701 (2003).Google Scholar
  53. 53. K.-I. Goh, B. Kahng, and D. Kim, Physica A 318, 72 (2003).Google Scholar
  54. 54. K.-I. Goh, B. Kahng and D. Kim, Phys. Rev. E 67, 017101 (2003).Google Scholar
  55. 55. J.-H. Kim, K.-I. Goh, B. Kahng, and D. Kim, Phys. Rev. Lett. 91, 058701 (2003).Google Scholar

Authors and Affiliations

  • Kwang-Il Goh
    • 1
  • Eulsik Oh
    • 1
  • Chul-Min Ghim
    • 1
  • Byungnam Kahng
    • 1
  • Doochul Kim
    • 1
  1. 1.School of Physics, Seoul National University, Seoul 151-747Korea

Personalised recommendations