[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Discontinuity, Nonlinearity and Complexity
ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Heavy-tailed Distributions in Some Stochastic Dynamical Models

Discontinuity, Nonlinearity and Complexity 1(1) 2012 1–40 | DOI:10.5890/DNC.2012.02.001

Ph. Blanchard, T. Kruger, D. Volchenkov$^1$

$^1$ Center of Excellence Cognitive Interaction Technology, Universit¨at Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany

Download Full Text PDF

 

Abstract

Heavy-tailed distributions are found throughout many naturally occurring phenomena. We have reviewed some models of stochastic dynamics that lead to heavy-tailed distributions (and power law distributions, in particular) including the multiplicative noise models, the models subjected to the Degree-Mass-Action principle (the generalized preferential attachment principle), the intermittent behavior occurring in complex physical systems near a bifurcation point, queuing systems, and the models of Self-organized criticality. Heavy-tailed distributions appear in them as the emergent phenomena sensitive for coupling rules essential for the entire dynamics.

Acknowledgments

We would like to thank Bruno Cessac, Elena Floriani, Max Hongler, and Ricardo Lima for numerous discussions.

References

  1. [1] Katz, Rw. ( 2002), ”Techniques for estimating uncertainty in climate change scenarios and impact studies”, Climate Research Vol. 20 (2), Pages: 167–185, Publisher: National Center for Atmospheric Research.
  2. [2] Wei-qi Liu (2008), ”The M-bootstrap estimation of heavy-tailed index and empirical analysis of Chinese stock markets”, Management Science and Engineering, ICMSE 2008. 15th Annual Conference Proceedings, 1275– 1285. (2008).
  3. [3] Levy, M., Solomon, S. (1996), Int. J. Mod. Phys. C, 7(4), 595–601.
  4. [4] Embrechts, P., Kl¨uppelberg, C., Mikosch, Th. (1996), Modelling Extremal Events for Insurance and Finance, Springer, Berlin.
  5. [5] Barab´asi, A.-L., Albert, R. (1999), Science, 286, 509–512.
  6. [6] Takayasu, H., Sato, A.-H., Takayasu, M. (1997), Phys. Rev. Lett., 79, 966– 969.
  7. [7] Albert, R., Barab´asi, A.-L. (2002), Rev. Mod. Phys., 74 (1).
  8. [8] L¨osch, A. (1954), The Economics of Location, Yale University Press, New Haven.
  9. [9] Henderson, J. (1974), Amer. Econ, Rev., LXIV, 640.
  10. [10] Floriani, E., Volchenkov, D., Lima, R. (2003), J. Phys. A: Math. Gen., 36, 4771-4783.
  11. [11] Pomeau, Y., Manneville, P. (1980), Commun. Math. Phys., 74, 1889.
  12. [12] Grebogi, C., Ott, E.,& Yorke, J.A. (1983),Phys. Rev. Lett., 50, 93.
  13. [13] Platt, N., Spiegel, E.A. , & Tresser, C. (1993), Phys. Rev. Lett., 70, 279.
  14. [14] Pikovsky, A.S. (1984), Z. Phys. B, 55, 149 .
  15. [15] Yu, L., Ott, E., Chen,Q. (1990), Phys. Rev. Lett., 65, 2935.
  16. [16] Venkataramani, S.C., Antonsen Jr., T.M., Ott, E., Sommerer, J.C. (1995), Phys. Lett. A, 207, 173.
  17. [17] Venkataramani, S.C., Antonsen Jr.,T.M., Ott, E., Sommerer, J.C. (1996), Physica D, 96, 66.
  18. [18] Hammer, P.W.,Platt, N., Hammel, S.M., Heagy, J.F., Lee, B.D. (1994), Phys. Rev. Lett., 73, 1095.
  19. [19] Zumofen, G., Klafter, J. (1993), Physica D, 69, 436.
  20. [20] Redondo, J., Rold´an, E., de Valc´arcel, G.J. (1996), Phys. Lett. A, 210, 301.
  21. [21] Heagy, J.F., Platt, N., Hammel, S.M. (1994), Phys. Rev. E, 49, 1140.
  22. [22] Platt, N., Hammel,S.M., Heagy, J.F. (1994), Phys. Rev. Lett., 72, 3498.
  23. [23] Qu,Z., Xie,F., Hu, G. (1996),Phys. Rev. E, 53, R1301.
  24. [24] Yang, H.L., Huang,Z.Q., Ding, E.J. (1996),Phys. Rev. E, 54, 3531.
  25. [25] Pierre, Th., Klostermann, H., Floriani, E., Lima, R. (2000), Phys. Rev. E 62, 7241.
  26. [26] Meiss, J.D. (1986), Phys. Rev. A, 34, 2375.
  27. [27] Newman, M.E.J., Sneppen, K. (1996), Phys. Rev. E, 54, 6226.
  28. [28] Sneppen, K., Newman, M.E.J. (1997), Physica D, 110, 209.
  29. [29] Bak, P., Tang, C., Wiesenfeld, K. (1987), Phys. Rev. Lett., 59, 381.
  30. [30] Wang, X.J. (1989), Phys. Rev. A, 40, 6647.
  31. [31] Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G. (1954), Tables of Integral Transforms, 2, McGraw-Hill, New York.
  32. [32] Flajolet, Ph., Sedgewick,R. (2009), Analytic Combinatorics, Cambridge Unviversity Press, Cambridge.
  33. [33] Blanchard, Ph., Cessac, B., Kr¨uger, T., 2000, J. Stat. Phys., 98, 375.
  34. [34] Baldwin, R.O., Davis, N.J., Kobza, J.E., Mikdiff ,S.F. (2000), Queueing Syst. Th. and Appl., 35(1-4), 1-21.
  35. [35] Barab´asi, A.-L. (2005), Nature, 435, 207.
  36. [36] Blanchard, Ph., Hongler,M.-O. (2004), Phys. Lett., A 323 (1-2), 63-66.
  37. [37] Blanchard, Ph., Hongler,M.-O. (2007), Phys. Rev. E, 75, 026102.
  38. [38] Boxma, O.J., Foss, S.G., Lasgouttes, J.M., N`u˜nez Queija, R. (2004), Queueing Systems, 46, 35-73.
  39. [39] Brauer & C, F., Castillo-Ch´avez (2001),MathematicalModels in Population Biology and Epidemiology”, in Appl. Math., 40, Springer.
  40. [40] Cohen. J.(1973), J. Appl. Probab., 10, 343-353 (1973).
  41. [41] Doytchinov, B., Lehozcky,J., Shreve, S. (2001), Ann. Appl. Probab., 11, 332-378.
  42. [42] Dusonchet, F., Hongler,M.-O. (2003) IEEE Trans Robot and Autom., 19(6), 997-990.
  43. [43] Filliger, R., Hongler,M.-O.(2005), Europhys. Lett., 70(3), 285-291.
  44. [44] Flatto, L. (1977), The Annals of Probab., 7, (2), 382-409.
  45. [45] Lehozcky, J.(1996), Real-time queuing theory, in Proc. IEEE Real-time system symposium, New-York, 186-195.
  46. [46] Lehozcky, J. (1997), Perfom. Eval., 25, 158-168.
  47. [47] Pakes, A.G. (1975), J. Appl. Prob., 12, 555-564.
  48. [48] Panwar, S., Townsley, D., Wolf, J.K. (1988), J. of the ACM, 35,(4), 832-844.
  49. [49] V´azquez, A. (2005), Phys. Rev. Lett., 95, 248701.
  50. [50] V´azquez, A., Oliveira, J.G., Dezs¨o, Z., Goh K.-I., Kondor, I. A.-L., Barab´asi (2006), Phys. Rev. E, 73(3), 036127.
  51. [51] Zhang, Y.-C. (1989),Phys. Rev. Lett., 63, 470.
  52. [52] Bak, P. (1996), How nature works, Springer-Verlag.
  53. [53] Jensen, H.J. ( 1998), Self-Organized Criticality : Emergent Complex Behavior in Physical and Biological systems, Cambridge Lecture Notes in Physics 10, Cambridge University Press.
  54. [54] Sornette, D., Johansen, A., Dornic, I. ( 1995), J. Phys. I France, 5, 325.
  55. [55] Bagnoli, F., Palmerini, P., Rechtman, R. ( 1997), Phys. Rev. E, 55, 3970.
  56. [56] Ma, S.K. (1976), Modern Theory of Critical Phenomena, Benjamin Reading.
  57. [57] Pietronero, L., Vespignani, A., Zapperi, S. (1994), Phys. Rev. Lett., 72, 1690.
  58. [58] Vespignani, A., Zapperi, S., Pietronero, L. (1995), Phys. Rev. E, 51, 1711.
  59. [59] Corral, A., D´ıaz-Guilera, A. (1997),Phys. Rev. E, 55 (3), 2434.
  60. [60] Ivashkevich E.V., Povolotsky, A.M., Vespignani, A., Zapperi, S. (1999), Phys. Rev. E, 60, 1239–1251.
  61. [61] Volchenkov, D., Cessac, B., Blanchard, Ph. (2002), Int. J. Mod. Phys. B, 16 (8), 1171.
  62. [62] Le Doussal, P., Wiese, K.J. (2009), Phys. Rev. E, 79, 051106.
  63. [63] Volchenkov, D. (2009), Eur. Phys. J. Special Topics, 170 (1), 1-142.
  64. [64] Foster, D., Nelson, D.R., Stephen, M.J. (1977),Phys. Rev. A, 16, 732.
  65. [65] Kadanoff, L.P., Nagel, S.R., Wu, L., Zhu, S. (1989),Phys. Rev. A, 39, 6524.
  66. [66] Grassberger, P., Manna, S.S. (1990), J. Phys. France, 51, 1077.
  67. [67] Cardy, J.L. (1984), J. Phys. A 17, L961.
  68. [68] Cardy J.L.(ed.) (1988), Finite-size Scaling, North-Holland, New York.
  69. [69] Pastor-Satorras, R., Vespignani, A.(2000), Eur. Phys. J. B, 18, 197-200.
  70. [70] De Menech, M., Stella, A.L. (2002), Physica A, 309 (3-4), 289-296 .
  71. [71] Tebaldi, C., De Menech, M., Stella, A. (1999), Phys. Rev. Lett., 83, 3952.
  72. [72] Hwa,T., Kadar, M., 1989,Phys. Rev. Lett., 62, 1813.
  73. [73] Grinstein, G., Lee, D.-H., Sachdev, S. (1990), Phys. Rev. Lett., 64, 1927.
  74. [74] D´ıaz-Guilera, A. (1994), Europhys. Lett., 26, 177.
  75. [75] P´erez, C.J., Corral, A., Diaz-Guilera A., Christensen, K., Arenas, A. (1996), Int. J. Mod. Phys. B, 10(10), 1111- 1151.
  76. [76] Collins, J. (1992), Renormalization: An Introduction to Renormalization, the Renormalization Group, and the Operator -Product Expansion, Cambridge University Press, Cambridge.
  77. [77] Martin, P.C., Siggia, E.D. and Rose, H.A. (1973), Phys. Rev. A, 8, 423.
  78. [78] de Dominicis, C., Peliti, L. (1978), Phys. Rev. B, 18, 353.
  79. [79] Zinn-Justin, J. (1990), Quantum Field Theory and Critical Phenomena, Clarendon, Oxford.
  80. [80] Wyld, H.W.(1961), Ann. Phys., 14, 143.
  81. [81] Martin, P.C., Siggia, E.D., Rose, H.A.(1973), Phys. Rev. A, 8, 423.
  82. [82] Monin, A.S., Yaglom, A.M.(1971),Statistical Fluid Mechanics, 1,2, MIT Press, Cambridge, Mass. (1971, 1975).
  83. [83] Pietronero, L., Tartagila, P., Zhang, Y.-C.(1991), Physica A, 173, 22.
  84. [84] Cessac, B., Blanchard, Ph., Kr¨uger, T.(2001),Phys. Rev. E, 64, 016133.
  85. [85] Adzhemyan, L.Ts., Antonov, N.V., Vasiliev A.N.(1998), Field Theoretic Renormalization Group in Fully Developed Turbulence, Gordon and Breach.
  86. [86] De Dominicis, C., Martin, P.C.(1979),Phys. Rev. A, 19, 419.
  87. [87] Antonov, N.V.(1999),Phys. Rev. E, 60, 6691.
  88. [88] Antonov, N.V.(1995), Sov. Phys. JTEP, 112, (11), 1649.
  89. [89] Antonov, N.V., Vasiliev, A.N. ( 19950, Sov. Phys. JETP, 108, 885.
  90. [90] Br´ezin, E., Leibler, S.(1982), Phys. Rev. B, 27, 594.
  91. [91] Kupiainen, A., Muratore–Ginanneschi, P. (2007), J. Stat. Phys., 126, 669-724.

Copyright © 2011-2023 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates