Table of Contents Author Guidelines Submit a Manuscript
The Scientific World Journal
Volume 2014, Article ID 759391, 8 pages
http://dx.doi.org/10.1155/2014/759391
Research Article

Geometric Assortative Growth Model for Small-World Networks

1Singapore University of Technology and Design, Singapore 138682
2Institute for Cyber Security, University of Texas at San Antonio, TX 78249, USA

Received 8 August 2013; Accepted 21 October 2013; Published 23 January 2014

Academic Editors: H. M. Chamberlin and Y. Zhang

Copyright © 2014 Yilun Shang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. D. J. Watts and S. H. Strogatz, “Collective dynamics of “small-world” networks,” Nature, vol. 393, no. 6684, pp. 440–442, 1998. View at Google Scholar · View at Scopus
  2. L. A. N. Amara, A. Scala, M. Barthélémy, and H. E. Stanley, “Classes of small-world networks,” Proceedings of the National Academy of Sciences of the United States of America, vol. 97, no. 21, pp. 11149–11152, 2000. View at Publisher · View at Google Scholar
  3. S. Jespersen and A. Blumen, “Small-world networks: links with long-tailed distributions,” Physical Review E, vol. 62, no. 5, pp. 6270–6274, 2000. View at Google Scholar · View at Scopus
  4. M. Kuperman and G. Abramson, “Complex structures in generalized small worlds,” Physical Review E, vol. 64, no. 4, Article ID 047103, 4 pages, 2001. View at Publisher · View at Google Scholar · View at Scopus
  5. M. E. J. Newman, “Models of the small world,” Journal of Statistical Physics, vol. 101, no. 3-4, pp. 819–841, 2000. View at Google Scholar · View at Scopus
  6. Y. Shang, “Fast distributed consensus seeking in large-scale sensor networks via shortcuts,” International Journal of Computational Science and Engineering, vol. 7, no. 2, pp. 121–124, 2012. View at Publisher · View at Google Scholar
  7. Y. Shang, “A sharp threshold for rainbow connection in small-world networks,” Miskolc Mathematical Notes, vol. 13, no. 2, pp. 493–497, 2012. View at Google Scholar
  8. F. Comellas and M. Sampels, “Deterministic small-world networks,” Physica A, vol. 309, no. 1-2, pp. 231–235, 2002. View at Publisher · View at Google Scholar · View at Scopus
  9. W. Xiao and B. Parhami, “Cayley graphs as models of deterministic small-world networks,” Information Processing Letters, vol. 97, no. 3, pp. 115–117, 2006. View at Publisher · View at Google Scholar · View at Scopus
  10. S. Boettcher, B. Gonçalves, and H. Guclu, “Hierarchical regular small-world networks,” Journal of Physics A, vol. 41, no. 25, Article ID 252001, 2008. View at Publisher · View at Google Scholar · View at Scopus
  11. Y. Zhang, Z. Zhang, S. Zhou, and J. Guan, “Deterministic weighted scale-free small-world networks,” Physica A, vol. 389, no. 16, pp. 3316–3324, 2010. View at Publisher · View at Google Scholar · View at Scopus
  12. M. E. J. Newman, “Assortative mixing in networks,” Physical Review Letters, vol. 89, no. 20, Article ID 208701, 4 pages, 2002. View at Publisher · View at Google Scholar
  13. M. E. J. Newman, “Mixing patterns in networks,” Physical Review E, vol. 67, no. 2, Article ID 026126, 13 pages, 2003. View at Publisher · View at Google Scholar
  14. Z. Zhang and F. Comellas, “Farey graphs as models for complex networks,” Theoretical Computer Science, vol. 412, no. 8–10, pp. 865–875, 2011. View at Publisher · View at Google Scholar · View at Scopus
  15. C. J. Colbourn, “Farey series and maximal outerplanar graphs,” SIAM Journal on Algebraic Discrete Methods, vol. 3, no. 2, pp. 187–189, 1982. View at Publisher · View at Google Scholar
  16. Z. Zhang, S. Zhou, Z. Wang, and Z. Shen, “A geometric growth model interpolating between regular and small-world networks,” Journal of Physics A, vol. 40, no. 39, pp. 11863–11876, 2007. View at Publisher · View at Google Scholar · View at Scopus
  17. Y. Shang, “Distinct clusterings and characteristic path lengths in dynamic smallworld networks with identical limit degree distribution,” Journal of Statistical Physics, vol. 149, no. 3, pp. 505–518, 2012. View at Publisher · View at Google Scholar
  18. S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, “Pseudofractal scale-free web,” Physical Review E, vol. 65, no. 6, Article ID 066122, 4 pages, 2002. View at Publisher · View at Google Scholar · View at Scopus
  19. Y. Shang, “Mean commute time for random walks on hierarchical scale-free networks,” Internet Mathematics, vol. 8, no. 4, pp. 321–337, 2012. View at Publisher · View at Google Scholar
  20. R. Albert and A. L. Barabási, “Statistical mechanics of complex networks,” Reviews of Modern Physics, vol. 74, no. 1, pp. 47–97, 2002. View at Publisher · View at Google Scholar
  21. M. E. J. Newman, “The structure and function of complex networks,” SIAM Review, vol. 45, no. 2, pp. 167–256, 2003. View at Google Scholar · View at Scopus
  22. S. Jung, S. Kim, and B. Kahng, “Geometric fractal growth model for scale-free networks,” Physical Review E, vol. 65, no. 5, Article ID 056101, 6 pages, 2002. View at Publisher · View at Google Scholar · View at Scopus
  23. J. S. Andrade Jr., H. J. Herrmann, R. F. S. Andrade, and L. R. Da Silva, “Apollonian networks: simultaneously scale-free, small world, euclidean, space filling, and with matching graphs,” Physical Review Letters, vol. 94, no. 1, Article ID 018702, 4 pages, 2009. View at Publisher · View at Google Scholar
  24. A.-L. Barabási, E. Ravasz, and T. Vicsek, “Deterministic scale-free networks,” Physica A, vol. 299, no. 3-4, pp. 559–564, 2001. View at Publisher · View at Google Scholar · View at Scopus
  25. A. Barrat, M. Barthélemy, R. Pastor-Satorras, and A. Vespignani, “The architecture of complex weighted networks,” Proceedings of the National Academy of Sciences of the United States of America, vol. 101, no. 11, pp. 3747–3752, 2004. View at Publisher · View at Google Scholar · View at Scopus
  26. R. Pastor-Satorras, A. Vázquez, and A. Vespignani, “Dynamical and correlation properties of the internet,” Physical Review Letters, vol. 87, no. 25, Article ID 258701, 4 pages, 2001. View at Publisher · View at Google Scholar · View at Scopus
  27. W. F. de la Vega and Z. Tuza, “Groupies in random graphs,” Information Processing Letters, vol. 109, no. 7, pp. 339–340, 2009. View at Publisher · View at Google Scholar · View at Scopus
  28. Y. Shang, “Groupies in random bipartite graphs,” Applicable Analysis and Discrete Mathematics, vol. 4, no. 2, pp. 278–283, 2010. View at Publisher · View at Google Scholar · View at Scopus
  29. H. Bateman and A. Erdelyi, Higher Transcendental Functions, vol. 1, McGraw-Hill, New York, NY, USA, 1953.
  30. M. E. J. Newman, D. J. Watts, and S. H. Strogatz, “Random graph models of social networks,” Proceedings of the National Academy of Sciences of the United States of America, vol. 99, no. 1, pp. 2566–2572, 2002. View at Publisher · View at Google Scholar · View at Scopus
  31. H. Ebel, L.-I. Mielsch, and S. Bornholdt, “Scale-free topology of e-mail networks,” Physical Review E, vol. 66, no. 3, Article ID 035103, 4 pages, 2002. View at Publisher · View at Google Scholar · View at Scopus
  32. Y. Shang, “Lack of Gromov-hyperbolicity in small-world networks,” Central European Journal of Mathematics, vol. 10, no. 3, pp. 1152–1158, 2012. View at Publisher · View at Google Scholar · View at Scopus
  33. M. Babaei, H. Ghassemieh, and M. Jalili, “Cascading failure tolerance of modular small-world networks,” IEEE Transactions on Circuits and Systems II, vol. 58, no. 8, pp. 527–531, 2011. View at Publisher · View at Google Scholar · View at Scopus