Random walks in modular scale-free networks with multiple traps

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 85, Issue 1, 2012, Pages

  Zhang, Zhongzhi ^a   Yang, Yihang ^a   Lin, Yuan ^a  

^a FUDAN UNIVERSITY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

ANALYTICAL RESULTS; COMPLEX RANDOM NETWORKS; COMPUTATIONAL APPROACH; DYNAMICAL PROCESS; EMPIRICAL INVESTIGATION; EXPECTED TIME; GENERAL NETWORKS; MEAN FIRST-PASSAGE TIME; MODULAR STRUCTURES; MULTIPLE TRAPS; POWER-LAW FUNCTIONS; RANDOM WALK; REAL NETWORKS; SCALE FREE NETWORKS; SCALE-FREE; SCALE-FREE TOPOLOGIES;

RANDOM PROCESSES;

BEHAVIORAL RESEARCH;

EID: 84863044344     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.85.011106     Document Type: Article

References (95)

1. R. Albert and A.-L. Barabási, Rev. Mod. Phys. RMPHAT 0034-6861 10.1103/RevModPhys.74.47 74, 47 (2002).
2. S. N. Dorogovtsev and J. F. F. Mendes, Adv. Phys. ADPHAH 0001-8732 10.1080/00018730110112519 51, 1079 (2002). (Pubitemid 34759032)
3. M. E. J. Newman, SIAM Rev. SIREAD 0036-1445 10.1137/S003614450342480 45, 167 (2003).
4. S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.-U. Hwanga, Phys. Rep. PRPLCM 0370-1573 10.1016/j.physrep.2005.10.009 424, 175 (2006). (Pubitemid 43135890)
5. A.-L. Barabási and R. Albert, Science SCIEAS 0036-8075 10.1126/science.286.5439.509 286, 509 (1999).
6. M. Girvan and M. E. J. Newman, Proc. Natl. Acad. Sci. USA PNASA6 0027-8424 10.1073/pnas.122653799 99, 7821 (2002). (Pubitemid 34650963)
7. E. Ravasz, A. L. Somera, D. A. Mongru, Z. N. Oltvai, and A.-L. Barabási, Science SCIEAS 0036-8075 10.1126/science.1073374 297, 1551 (2002). (Pubitemid 34971002)
8. E. Ravasz and A.-L. Barabási, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.67.026112 67, 026112 (2003).
9. F. Chung and L. Lu, Proc. Natl. Acad. Sci. USA PNASA6 0027-8424 10.1073/pnas.252631999 99, 15879 (2002). (Pubitemid 35462716)
10. R. Cohen and S. Havlin, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.90.058701 90, 058701 (2003).
11. L. Danon, J. Duch, A. Diaz-Guilera, and A. Arenas, J. Stat. Mech.: Theory Exp. PRLTAO 1742-5468 10.1088/1742-5468/2005/09/P09008 (2005) P09008.
12. G. Palla, I. Derényi, I. Farkas, and T. Vicsek, Nature (London) NATUAS 0028-0836 10.1038/nature03607 435, 814 (2005). (Pubitemid 40839731)
13. M. E. J. Newman, Proc. Natl. Acad. Sci. USA PNASA6 0027-8424 10.1073/pnas.0601602103 103, 8577 (2006). (Pubitemid 43878062)
14. S. Fortunato, Phys. Rep. PRPLCM 0370-1573 10.1016/j.physrep.2009.11.002 486, 75 (2010).
15. W. H. Zhan, Z. Z. Zhang, J. H. Guan, and S. G. Zhou, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.83.066120 83, 066120 (2011).
16. C. E. Perkins, Ad Hoc Networking, 1st ed. (Addison-Wesley, Reading, MA, 2001).
17. A. Y. Wu, M. Garland, and J. Han, in KDD'04: Proceedings of the Tenth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (ACM, New York, 2004), pp. 719-724.
18. S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, Rev. Mod. Phys. RMPHAT 0034-6861 10.1103/RevModPhys.80.1275 80, 1275 (2008).
19. S. Havlin and D. ben-Avraham, Adv. Phys. ADPHAH 0001-8732 10.1080/00018738700101072 36, 695 (1987).
20. R. Metzler and J. Klafter, Phys. Rep. PRPLCM 0370-1573 10.1016/S0370-1573(00)00070-3 339, 1 (2000).
21. R. Metzler and J. Klafter, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/37/31/R01 37, R161 (2004). (Pubitemid 39063557)
22. R. Burioni and D. Cassi, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/ 38/8/R01 38, R45 (2005). (Pubitemid 40306586)
23. I. M. Sokolov, J. Mai, and A. Blumen, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.79.857 79, 857 (1997).
24. S. A. Pandit and R. E. Amritkar, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.63.041104 63, 041104 (2001).
25. J. D. Noh and H. Rieger, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.92.118701 92, 118701 (2004).
26. O. Bénichou, M. Coppey, M. Moreau, P.-H. Suet, and R. Voituriez, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.94.198101 94, 198101 (2005). (Pubitemid 41493011)
27. V. Sood, S. Redner, and D. ben-Avraham, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/38/1/007 38, 109 (2005). (Pubitemid 40111486)
28. E. M. Bollt and D. ben-Avraham, New J. Phys. NJOPFM 1367-2630 10.1088/1367-2630/7/1/026 7, 26 (2005). (Pubitemid 41093488)
29. L. K. Gallos, C. Song, S. Havlin, and H. A. Makse, Proc. Natl. Acad. Sci. USA PNASA6 0027-8424 10.1073/pnas.0700250104 104, 7746 (2007). (Pubitemid 47185822)
30. P. Barthelemy, J. Bertolotti, and D. S. Wiersma, Nature (London) NATUAS 0028-0836 10.1038/nature06948 453, 495 (2008). (Pubitemid 351733320)
31. A. Baronchelli, M. Catanzaro, and R. Pastor-Satorras, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.78.011114 78, 011114 (2008).
32. A. Kittas, S. Carmi, S. Havlin, and P. Argyrakis, Europhys. Lett. EULEEJ 0295-5075 10.1209/0295-5075/84/40008 84, 40008 (2008).
33. Z. Z. Zhang, Y. C. Zhang, S. G. Zhou, M. Yin, and J. H. Guan, J. Math. Phys. JMAPAQ 0022-2488 10.1063/1.3094757 50, 033514 (2009).
34. C. P. Haynes and A. P. Roberts, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.103.020601 103, 020601 (2009).
35. Z. Z. Zhang, J. L. T. Alafate, B. Y. Hou, H. J. Zhang, and G. R. Chen, Eur. Phys. J. B 10.1140/epjb/e2011-20834-1 84, 691 (2011).
36. G. H. Weiss, Aspects and Applications of the Random Walk (North-Holland, Amsterdam, 1994).
37. B. H. Hughes, Random Walks and Random Environments, Vols. 1 and 2 (Clarendon, Oxford, 1995).
38. F. Jasch and A. Blumen, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.63.041108 63, 041108 (2001).
39. M. F. Shlesinger, Nature (London) NATUAS 0028-0836 10.1038/443281a 443, 281 (2006). (Pubitemid 44435133)
40. J. Whitmarsh and J. A. Govindjee, in Concepts in Photobiology: Photosynthesis and Photomorphogenesis, edited by, G. Singhal, G. Renger, S. Sopory, K.-D. Irrgang, and, J. A. Govindjee, (Narosa, New Delhi, 1999), pp. 11-51.
41. F. Fouss, A. Pirotte, J. M. Renders, and M. Saerens, IEEE Trans. Knowl. Data Eng. ITKEEH 1041-4347 10.1109/TKDE.2007.46 19, 355 (2007). (Pubitemid 46374533)
42. E. W. Montroll, J. Math. Phys. JMAPAQ 0022-2488 10.1063/1.1664902 10, 753 (1969).
43. J. J. Kozak and V. Balakrishnan, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.65.021105 65, 021105 (2002). (Pubitemid 40619364)
44. J. J. Kozak and V. Balakrishnan, Int. J. Bifurcation Chaos Appl. Sci. Eng. IJBEE4 0218-1274 10.1142/S0218127402006138 12, 2379 (2002). (Pubitemid 36156094)
45. Z. Z. Zhang, B. Wu, H. J. Zhang, S. G. Zhou, J. H. Guan, and Z. G. Wang, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.81.031118 81, 031118 (2010).
46. E. Agliari, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.77.011128 77, 011128 (2008). (Pubitemid 351205586)
47. Z. Z. Zhang, Y. Lin, S. G. Zhou, B. Wu, and J. H. Guan, New J. Phys. NJOPFM 1367-2630 10.1088/1367-2630/11/10/103043 11, 103043 (2009).
48. A. Garcia Cantú and E. Abad, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.77.031121 77, 031121 (2008). (Pubitemid 351439070)
49. Z. Z. Zhang, Y. Qi, S. G. Zhou, W. L. Xie, and J. H. Guan, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.79.021127 79, 021127 (2009).
50. Z. Z. Zhang, S. G. Zhou, W. L. Xie, L. C. Chen, Y. Lin, and J. H. Guan, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.79.061113 79, 061113 (2009).
51. Z. Z. Zhang, J. H. Guan, W. L. Xie, Y. Qi, and S. G. Zhou, Europhys. Lett. EULEEJ 0295-5075 10.1209/0295-5075/86/10006 86, 10006 (2009).
52. E. Agliari and R. Burioni, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.80.031125 80, 031125 (2009).
53. V. Tejedor, O. Bénichou, and R. Voituriez, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.80.065104 80, 065104 (R) (2009).
54. Z. Z. Zhang, W. L. Xie, S. G. Zhou, S. Y. Gao, and J. H. Guan, Europhys. Lett. EULEEJ 0295-5075 10.1209/0295-5075/88/10001 88, 10001 (2009).
55. Z. Z. Zhang, W. L. Xie, S. G. Zhou, M. Li, and J. H. Guan, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.80.061111 80, 061111 (2009).
56. S. Condamin, O. Bénichou, V. Tejedor, R. Voituriez, and J. Klafter, Nature (London) NATUAS 0028-0836 10.1038/nature06201 450, 77 (2007). (Pubitemid 350070561)
57. O. Bénichou, B. Meyer, V. Tejedor, and R. Voituriez, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.101.130601 101, 130601 (2008).
58. S. Condamin, V. Tejedor, R. Voituriez, O. Bénichou, and J. Klafter, Proc. Natl. Acad. Sci. USA PNASA6 0027-8424 10.1073/pnas.0712158105 105, 5675 (2008). (Pubitemid 351758430)
59. S. B. Yuste and L. Acedo, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.68.036134 68, 036134 (2003).
60. T. A. Witten and L. M. Sander, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.47.1400 47, 1400 (1981).
61. T. A. Witten and L. M. Sander, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.27.5686 27, 5686 (1983).
62. S. A. Rice, in Diffusion-Limited Reactions, edited by, C. H. Bamford, C. F. H. Tipper, and, R. G. Compton, Comprehensive Chemical Kinetics, Vol. 25 (Elsevier, Amsterdam, 1985).
63. M. Hinczewski, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.75.061104 75, 061104 (2007). (Pubitemid 47140345)
64. S. Maslov, Nature Phys. 10.1038/nphys493 3, 18 (2007).
65. A. Diaz-Guilera, J. Phys. A PRLTAO 1751-8113 10.1088/1751-8113/41/22/ 224007 41, 224007 (2008).
66. Z. Z. Zhang, Y. Lin, S. Y. Gao, S. G. Zhou, J. H. Guan, and M. Li, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.80.051120 80, 051120 (2009).
67. E. Bullmore and O. Sporns, Nature Rev. Neurosci. PRLTAO 1471-0048 10.1038/nrn2575 10, 186 (2009).
68. S. Redner, A Guide to First-Passage Processes (Cambridge University Press, Cambridge, 2001).
69. S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.65.066122 65, 066122 (2002). (Pubitemid 40606656)
70. S. Jung, S. Kim, and B. Kahng, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.65.056101 65, 056101 (2002). (Pubitemid 40606838)
71. F. Comellas, J. Ozón, and J. G. Peters, Inf. Process. Lett. IFPLAT 0020-0190 10.1016/S0020-0190(00)00118-6 76, 83 (2000).
72. Z. Z. Zhang, L. L. Rong, and C. H. Guo, Physica A PHYADX 0378-4371 10.1016/j.physa.2005.08.020 363, 567 (2006). (Pubitemid 43375038)
73. M. Hinczewski and A. N. Berker, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.73.066126 73, 066126 (2006). (Pubitemid 43974164)
74. H. D. Rozenfeld, S. Havlin, and D. ben-Avraham, New J. Phys. NJOPFM 1367-2630 10.1088/1367-2630/9/6/175 9, 175 (2007). (Pubitemid 47146380)
75. Z. Z. Zhang, S. G. Zhou, T. Zou, L. C. Chen, and J. H. Guan, Eur. Phys. J. B EPJBFY 1434-6028 10.1140/epjb/e2007-00344-7 60, 259 (2007). (Pubitemid 350277669)
76. Z. Z. Zhang, S. G. Zhou, L. J. Fang, J. H. Guan, and Y. C. Zhang, Europhys. Lett. EULEEJ 0295-5075 10.1209/0295-5075/79/38007 79, 38007 (2007).
77. S. Boettcher, B. Gonçalves, and H. Guclu, J. Phys. A EULEEJ 1751-8113 10.1088/1751-8113/41/25/252001 41, 252001 (2008).
78. F. Comellas and A. Miralles, Physica A PHYADX 0378-4371 10.1016/j.physa.2009.02.004 388, 2227 (2009).
79. K. L. Morrow, T. Rowland, and C. M. Danforth, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.80.016103 80, 016103 (2009).
80. Z. Z. Zhang, J. H. Guan, B. L. Ding, L. C. Chen, and S. G. Zhou, New J. Phys. NJOPFM 1367-2630 10.1088/1367-2630/11/8/083007 11, 083007 (2009).
81. A. Miralles, F. Comellas, L. C. Chen, and Z. Z. Zhang, Physica A PHYADX 0378-4371 10.1016/j.physa.2009.12.056 389, 1955 (2010).
82. J. D. Noh, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.67.045103 67, 045103 (R) (2003).
83. J. D. Noh and H. Rieger, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.69.036111 69, 036111 (2004).
84. A.-L. Barabási, E. Ravasz, and T. Vicsek, Physica A PHYADX 0378-4371 10.1016/S0378-4371(01)00369-7 299, 559 (2001). (Pubitemid 32935251)
85. K. Iguchi and H. Yamada, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.71.036144 71, 036144 (2005). (Pubitemid 40629722)
86. Z. Z. Zhang, Y. Lin, S. Y. Gao, S. G. Zhou, and J. H. Guan, J. Stat. Mech.: Theory Exp. PHYADX 1742-5468 10.1088/1742-5468/2009/10/P10022 (2009) P10022.
87. E. Agliari, R. Burioni, and A. Manzotti, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.82.011118 82, 011118 (2010).
88. Z. Z. Zhang and Y. Lin, J. Stat. Mech.: Theory Exp. PHYADX 1742-5468 10.1088/1742-5468/2010/12/P12017 (2010) P12017.
89. D. J. Watts and H. Strogatz, Nature (London) NATUAS 0028-0836 10.1038/30918 393, 440 (1998). (Pubitemid 28292183)
90. J. G. Kemeny and J. L. Snell, Finite Markov Chains (Springer, New York, 1976).
91. D. Aldous and J. Fill http://www.stat.berkeley.edu/~aldous/RWG/Chap2.pdf.
92. A. Bar-Haim and J. Klafter, J. Chem. Phys. JCPSA6 0021-9606 10.1063/1.477135 109, 5187 (1998). (Pubitemid 128678689)
93. i is taken over all nodes other than the traps. However, when the concentration of traps (the ratio of the trap number Γ to network size N) is very small, both definitions lead to the same behavior of MFPT T.
94. H. S. Wilf, Generatingfunctionology, 2nd ed. (Academic, London, 1994).
95. V. Tejedor, O. Bénichou, and R. Voituriez, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.83.066102 83, 066102 (2011).