Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2017, Article ID 8217361, 14 pages
https://doi.org/10.1155/2017/8217361
Research Article

Mean First Passage Time of Preferential Random Walks on Complex Networks with Applications

1School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai 201620, China
2School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798
3School of Information Science and Technology, Donghua University, Shanghai 201620, China

Correspondence should be addressed to Zhongtuan Zheng; moc.361@gnehznautgnohz

Received 23 March 2017; Accepted 14 May 2017; Published 17 August 2017

Academic Editor: Huanqing Wang

Copyright © 2017 Zhongtuan Zheng et al. 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. S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.-U. Hwang, “Complex networks: structure and dynamics,” Physics Reports. A Review Section of Physics Letters, vol. 424, no. 4-5, pp. 175–308, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. M. E. Newman, “The structure and function of complex networks,” SIAM Review, vol. 45, no. 2, pp. 167–256, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. D. J. Watts and S. H. Strogatz, “Collective dynamics of 'small-world' networks,” Nature, vol. 393, no. 6684, pp. 440–442, 1998. View at Publisher · View at Google Scholar · View at Scopus
  4. A.-L. Barabási and R. Albert, “Emergence of scaling in random networks,” American Association for the Advancement of Science. Science, vol. 286, no. 5439, pp. 509–512, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  5. K. Li, Z. Ma, Z. Jia, M. Small, and X. Fu, “Interplay between collective behavior and spreading dynamics on complex networks,” Chaos. An Interdisciplinary Journal of Nonlinear Science, vol. 22, no. 4, Article ID 043113, 043113, 10 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. W.-X. Wang, B.-H. Wang, B. Hu, G. Yan, and Q. Ou, “General dynamics of topology and traffic on weighted technological networks,” Physical Review Letters, vol. 94, no. 18, Article ID 188702, 2005. View at Publisher · View at Google Scholar · View at Scopus
  7. Y. Li, S. Tong, and T. Li, “Observer-based adaptive fuzzy tracking control of MIMO stochastic nonlinear systems with unknown control direction and unknown dead-zones,” IEEE Transactions on Fuzzy Systems, vol. 23, no. 4, pp. 1228–1241, 2015. View at Publisher · View at Google Scholar
  8. X. Zhao, P. Shi, X. Zheng, and L. Zhang, “Adaptive tracking control for switched stochastic nonlinear systems with unknown actuator dead-zone,” Automatica. A Journal of IFAC, the International Federation of Automatic Control, vol. 60, pp. 193–200, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. Y. Li, S. Sui, and S. Tong, “Adaptive fuzzy control design for stochastic nonlinear switched systems with arbitrary switchings and unmodeled dynamics,” IEEE Transactions on Cybernetics, vol. 47, no. 2, pp. 403–414, 2017. View at Publisher · View at Google Scholar · View at Scopus
  10. H. Wang, W. Sun, and P. X. Liu, “Adaptive intelligent control of nonaffine nonlinear time-delay systems with dynamic uncertainties,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, no. 99, pp. 1–12, 2016. View at Google Scholar
  11. J. D. O. Noh and H. Rieger, “Random walks on complex networks,” Physical review letters, vol. 92, no. 11, p. 118701, 2004. View at Publisher · View at Google Scholar · View at Scopus
  12. R. Durrett, Random graph dynamics, Cambridge University Press, Cambridge, UK, 2007.
  13. S. A. Pandit and R. E. Amritkar, “Random spread on the family of small-world networks,” Physical Review E, vol. 63, no. 4, 2001. View at Publisher · View at Google Scholar
  14. S. Jespersen, I. M. Sokolov, and A. Blumen, “Relaxation properties of small-world networks,” Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, vol. 62, no. 3 B, pp. 4405–4408, 2000. View at Publisher · View at Google Scholar · View at Scopus
  15. W. Sun, “Random walks on generalized Koch networks,” Physica Scripta, vol. 88, no. 4, Article ID 045006, 2013. View at Publisher · View at Google Scholar
  16. J. D. Noh and S. Kim, “Random-walk and pair-annihilation processes on scale-free networks,” Journal of the Korean Physical Society, vol. 48, pp. 202–207, 2006. View at Google Scholar
  17. Z. Zheng, H. Wang, S. Gao, and G. Wang, “Comparison of multiple random walks strategies for searching networks,” Mathematical Problems in Engineering, vol. 2013, Article ID 734630, 2013. View at Publisher · View at Google Scholar · View at Scopus
  18. B. J. Kim, C. N. Yoon, S. K. Han, and H. Jeong, “Path finding strategies in scale-free networks,” Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, vol. 65, no. 2, Article ID 027103, p. 027103/4, 2002. View at Publisher · View at Google Scholar · View at Scopus
  19. N. Perra, A. Baronchelli, D. Mocanu, B. Gonçalves, R. Pastor-Satorras, and A. Vespignani, “Random walks and search in time-varying networks,” Physical Review Letters, vol. 109, no. 23, Article ID 238701, 2012. View at Publisher · View at Google Scholar · View at Scopus
  20. S.-P. Wang and W.-J. Pei, “Detecting unknown paths on complex networks through random walks,” Physica A: Statistical Mechanics and its Applications, vol. 388, no. 4, pp. 514–522, 2009. View at Publisher · View at Google Scholar · View at Scopus
  21. H. Tian, H. Shen, and T. Matsuzawa, “Random walk routing for wireless sensor networks,” in Proceedings of the 6th International Conference on Parallel and Distributed Computing, Applications and Technologies, PDCAT 2005, pp. 196–200, chn, December 2005. View at Publisher · View at Google Scholar · View at Scopus
  22. V. M. López Millán, V. Cholvi, L. López, and A. Fernández Anta, “A model of self-avoiding random walks for searching complex networks,” Networks, vol. 60, no. 2, pp. 71–85, 2012. View at Publisher · View at Google Scholar · View at Scopus
  23. C.-M. Angelopoulos, S. Nikoletseas, D. Patroumpa, and J. Rolim, “Coverage-adaptive random walks for fast sensory data collection,” Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 6288, pp. 81–94, 2010. View at Publisher · View at Google Scholar · View at Scopus
  24. A. Fronczak and P. Fronczak, “Biased random walks in complex networks: The role of local navigation rules,” Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, vol. 80, no. 1, Article ID 016107, 2009. View at Publisher · View at Google Scholar · View at Scopus
  25. S. Lee, S.-H. Yook, and Y. Kim, “Searching method through biased random walks on complex networks,” Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, vol. 80, no. 1, Article ID 017102, 2009. View at Publisher · View at Google Scholar · View at Scopus
  26. A. P. Riascos and J. L. Mateos, “Long-range navigation on complex networks using Lévy random walks,” Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, vol. 86, no. 5, Article ID 056110, 2012. View at Publisher · View at Google Scholar · View at Scopus
  27. Z. Zhang and S. Gao, “Scaling of mean first-passage time as efficiency measure of nodes sending information on scale-free Koch networks,” European Physical Journal B, vol. 80, no. 2, pp. 209–216, 2011. View at Publisher · View at Google Scholar · View at Scopus
  28. S. Hwang, D.-S. Lee, and B. Kahng, “First passage time for random walks in heterogeneous networks,” Physical Review Letters, vol. 109, no. 8, Article ID 088701, 2012. View at Publisher · View at Google Scholar · View at Scopus
  29. S. Condamin, O. Bénichou, V. Tejedor, R. Voituriez, and J. Klafter, “First-passage times in complex scale-invariant media,” Nature, vol. 450, no. 7166, pp. 77–80, 2007. View at Publisher · View at Google Scholar · View at Scopus
  30. V. Tejedor, O. Bénichou, and R. Voituriez, “Close or connected: Distance and connectivity effects on transport in networks,” Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, vol. 83, no. 6, Article ID 066102, 2011. View at Publisher · View at Google Scholar · View at Scopus
  31. Y. Lin, A. Julaiti, and Z. Zhang, “Mean first-passage time for random walks in general graphs with a deep trap,” Journal of Chemical Physics, vol. 137, no. 12, Article ID 124104, 2012. View at Publisher · View at Google Scholar · View at Scopus
  32. O. Häggström, Finite Markov Chains And Algorithmic Applications, Cambridge University Press, Cambridge, UK, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  33. S. M. Ross, Stochastic Processes, John Wiley, New York, NY, USA, 1996. View at MathSciNet
  34. R. Durrett, Probability: Theory and Examples, Cambridge University Press, Cambridge, UK, 4th edition, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  35. Z. T. Zheng, Exploring complex networks by random graph evolution and random walks [Ph.D. Dissertation], Shanghai University, Shanghai, China, 2009.
  36. Z. Zhang, T. Shan, and G. Chen, “Random walks on weighted networks,” Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, vol. 87, no. 1, Article ID 012112, 2013. View at Publisher · View at Google Scholar · View at Scopus
  37. 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
  38. Y. Li, Z. Ma, and S. Tong, “Adaptive Fuzzy Output-Constrained Fault-Tolerant Control of Nonlinear Stochastic Large-Scale Systems With Actuator Faults,” IEEE Transactions on Cybernetics, vol. 47, no. 9, pp. 2362–2376, 2017. View at Publisher · View at Google Scholar
  39. H. Wang, P. X. Liu, and P. Shi, “Observer-based fuzzy adaptive output-feedback control of stochastic nonlinear multiple time-delay systems,” IEEE Transactions on Cybernetics, vol. 47, no. 9, pp. 2568–2578, 2017. View at Publisher · View at Google Scholar
  40. Y. M. Li and S. C. Tong, “Adaptive fuzzy output constrained control design for multi-input multioutput stochastic nonstrict-feedback nonlinear systems,” IEEE Transactions on Cybernetics, no. 99, pp. 1–10, 2016. View at Publisher · View at Google Scholar · View at Scopus
  41. X. Zhao, P. Shi, X. Zheng, and J. Zhang, “Intelligent tracking control for a class of uncertain high-order nonlinear systems,” IEEE Transactions on Neural Networks and Learning Systems, vol. 27, no. 9, pp. 1976–1982, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  42. L. Lovász, “Random walks on graphs: a survey,” in Combinatorics, Paul Erdös is eighty, vol. 2, pp. 1–46, János Bolyai Mathematical Society, Budapest, Hungary, 1993. View at Google Scholar
  43. C. M. Grinstead and J. L. Snell, Introduction to Probability, American Mathematical Society, Providence, RI, USA, 1997.
  44. D. J. Aldous and J. Fill, “Reversible Markov Chains and Random Walks on Graphs, (monograph in preparation),” http://www.stat.berkeley.edu/ãldous/RWG/book.html, 2002.
  45. F. R. Chung, Spectral Graph Theory, American Mathematical Society, Providence, RI, USA, 1997. View at MathSciNet
  46. J. M. Kleinberg, “Navigation in a small world,” Nature, vol. 406, no. 6798, p. 845, 2000. View at Publisher · View at Google Scholar · View at Scopus
  47. M. Kitsak, L. K. Gallos, S. Havlin et al., “Identification of influential spreaders in complex networks,” Nature Physics, vol. 6, no. 11, pp. 888–893, 2010. View at Publisher · View at Google Scholar · View at Scopus