Universality of Zipf's law

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

Volumn 82, Issue 1, 2010, Pages

  Corominas Murtra, Bernat ^a   Solé, Ricard V ^a,b,c  

^a UNIVERSITAT POMPEU FABRA   (Spain)

^b SANTA FE INSTITUTE   (United States)

^c INSTITUTE OF EVOLUTIONARY BIOLOGY CSIC UNIVERSITAT POMPEU FABRA   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMIC INFORMATION THEORY; GENERAL CLASS; GENERAL NATURE; MODEL FREE; ORDER AND DISORDER; REAL SYSTEMS; SCALING BEHAVIOR; STABLE STATE; STATISTICAL DISTRIBUTION; SYMBOLIC SEQUENCE; THEORETICAL EXPLANATION; ZIPF'S LAW;

INFORMATION THEORY;

EID: 77954322322     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.82.011102     Document Type: Article

References (69)

1. H. E. Stanley, L. A. N. Amaral, P. Gopikrishnan, P. Ch. Ivanov, T. H. Keitt, and V. Plerou, Physica A 281, 60 (2000). 10.1016/S0378-4371(00)00195-3
2. H. E. Stanley, Rev. Mod. Phys. 71, S358 (1999). 10.1103/RevModPhys.71. S358
3. M. E. J. Newman, Contemp. Phys. 46, 323 (2005). 10.1080/00107510500052444
4. R. V. Solé and B. Goodwin, Signs of Life: How Complexity Pervades Biology (Basic Books, New York, 2001).
5. G. K. Zipf, Human Behavior and the Principle of Least Effort (Addison-Wesley, Reading, MA, 1949).
6. F. Auerbach, Petermans Geograpische Mittelungen 59, 74 (1913).
7. X. Gabaix, Q. J. Econ. 114, 739 (1999). 10.1162/003355399556133
8. R. Ferrer i Cancho and R. V. Solé, Adv. Complex Syst. 5, 1 (2002). 10.1142/S0219525902000468
9. For a very complete collection of the works concerning the topic we refer the reader to: http://www.nslij-genetics.org/wli/zipf/
10. H. A. Simon, Biometrika 42, 425 (1955). 10.1093/biomet/42.3-4.425
11. H. A. Makse, S. Havlin, and H. E. Stanley, Nature (London) 377, 608 (1995). 10.1038/377608a0
12. P. Krugman, J. Jpn. Int. Econ. 10, 399 (1996). 10.1006/jjie.1996.0023
13. A. Blank and S. Solomon, Physica A 287, 279 (2000). 10.1016/S0378- 4371(00)00464-7
14. E. H. Decker, A. J. Kerkhoff, and M. E. Moses, PLoS ONE 2, e934 (2007). 10.1371/journal.pone.0000934
15. R. L. Axtell, Science 293, 1818 (2001). 10.1126/science.1062081
16. X. Gabaix, P. Gopikrishnan, V. Plerou, and E. H. Stanley, Nature (London) 423, 267 (2003). 10.1038/nature01624
17. V. Pareto, Cours d'Economie Politique (Droz, Geneva, 1896).
18. K. Okuyama, M. Takayasu, and H. Takayasu, Physica A 269, 125 (1999). 10.1016/S0378-4371(99)00086-2
19. W. J. Reed and B. D. Hughes, Phys. Rev. E 66, 067103 (2002). 10.1103/PhysRevE.66.067103
20. D. H. Zanette and S. C. Manrubia, Physica A 295, 1 (2001). 10.1016/S0378-4371(01)00046-2
21. P. Krugman, The Self-Organizing Economy (Blackwell, Oxford, 1996).
22. B. W. Arthur, Increasing Returns and Path Dependence in Economy (Michigan University Press, Michigan, 1996).
23. S. C. Manrubia, D. H. Zanette, and R. V. Solé, Fractals 7, 1 (1999). 10.1142/S0218348X99000025
24. G. Rozman, in East Asian Urbanization in the Nineteenth Century: Comparisons with Europe, edited by, V. der Woude,, (Clarendon, Oxford, 1990), pp. 61-63.
25. S. Solomon and M. Levy, Int. J. Mod. Phys. C 7, 745 (1996). 10.1142/S0129183196000624
26. O. Malcai, O. Biham, and S. Solomon, Phys. Rev. E 60, 1299 (1999). 10.1103/PhysRevE.60.1299
27. Z. F. Huang and S. Solomon, Eur. Phys. J. B 20, 601 (2001). 10.1007/PL00011114
28. R. V. Solé and S. C. Manrubia, Phys. Rev. E 54, R42 (1996). 10.1103/PhysRevE.54.R42
29. M. E. J. Newman and R. G. Palmer, Modelling Extinction (Oxford University Press, New York, 2003).
30. D. H. Zanette and S. C. Manrubia, Phys. Rev. Lett. 79, 523 (1997). 10.1103/PhysRevLett.79.523
31. S. C. Manrubia and D. H. Zanette, Phys. Rev. E 58, 295 (1998). 10.1103/PhysRevE.58.295
32. M. E. J. Newman and K. Sneppen, Phys. Rev. E 54, 6226 (1996). 10.1103/PhysRevE.54.6226
33. W. H. White, J. Colloid Interface Sci. 87, 204 (1982). 10.1016/0021-9797(82)90382-4
34. F. Family and P. Meakin, Phys. Rev. A 40, 3836 (1989). 10.1103/PhysRevA.40.3836
35. P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987). 10.1103/PhysRevLett.59.381
36. P. Harremoës and F. Topsøe, Entropy 3, 191 (2001). 10.3390/e3030191
37. R. Ferrer i Cancho and R. V. Solé, Proc. Natl. Acad. Sci. U.S.A. 100, 788 (2003). 10.1073/pnas.0335980100
38. W. Li, IEEE Trans. Inf. Theory 38, 1842 (1992). 10.1109/18.165464
39. E. W. Montroll and M. F. Shlesinger, Proc. Natl. Acad. Sci. U.S.A. 79, 3380 (1982). 10.1073/pnas.79.10.3380
40. K. Kawamura and N. Hatano, J. Phys. Soc. Jpn. 71, 1211 (2002). 10.1143/JPSJ.71.1211
41. Luis A. Nunes Amaral, S. V. Buldyrev, S. Havlin, M. A. Salinger, and H. E. Stanley, Phys. Rev. Lett. 80, 1385 (1998). 10.1103/PhysRevLett.80.1385
42. R. Solomonoff, Inform and Control 7, 1 (1964). 10.1016/S0019-9958(64) 90223-2
43. A. Kolmogorov, Probl. Inf. Transm. 1, 1 (1965).
44. G. J. Chaitin, J. ACM 13, 547 (1966). 10.1145/321356.321363
45. L. Ming and P. Vitányi, An Introduction to Kolmogorov Complexity and its Applications (Springer, New York, 1997).
46. R. Perline, Phys. Rev. E 54, 220 (1996). 10.1103/PhysRevE.54.220
47. T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991). 10.1002/0471200611
48. C. Adami, Introduction to Artificial Life (Springer, New York, 1999).
49. G. J. Chaitin, J. ACM 22, 329 (1975). 10.1145/321892.321894
50. J. S. Nicolis, Rep. Prog. Phys. 49, 1109 (1986). 10.1088/0034-4885/49/10/ 002
51. W. Ebeling and G. Nicolis, EPL 14, 191 (1991). 10.1209/0295-5075/14/3/001
52. W. Ebeling, Physica A 194, 563 (1993). 10.1016/0378-4371(93)90386-I
53. F. Kaspar and H. G. Schuster, Phys. Rev. A 36, 842 (1987). 10.1103/PhysRevA.36.842
54. W. H. Zurek, Nature (London) 341, 119 (1989). 10.1038/341119a0
55. T. G. Dewey, Phys. Rev. E 54, R39 (1996). 10.1103/PhysRevE.54.R39
56. P. D. Grünwald and P. M. Vitányi, in Handbook of the Philosophy of Science, Volume 8: Philosophy of Information, edited by, P. Adriaans, and, J. van Benthem, (Elsevier, New York, 2008), pp. 289-325.
57. C. E. Shannon, Bell Syst. Tech. J. 27, 379 (1948).
58. R. B. Ash, Information Theory (Dover, New York, 1990).
59. P. D. Grünwald and P. M. B. Vitányi, J. Log. Lang. Inf. 12, 497 (2003). 10.1023/A:1025011119492
60. E. T. Jaynes, Phys. Rev. 106, 620 (1957). 10.1103/PhysRev.106.620
61. H. Haken, Synergetics: An Introduction. Nonequilibrium Phase Transitions and Self- Organization in Physics, Chemistry and Biology, Springer Series in Synergetics (Springer, New York, 1978).
62. J. N. Kapur, Maximum-entropy Models in Science and Engineering (Wiley Eastern Limited, New Delhi, 1989).
63. R. K. Pathria, Statistical Mechanics, 2nd ed. (Butterworth-Heinemann, 1996).
64. A. Hernando, D. Puigdomènech, D. Villuendas, C. Vesperinas, and A. Plastino, Phys. Lett. A 374, 18 (2009). 10.1016/j.physleta.2009.10.027
65. Handbook of Mathematical Functions, NBS, Applied Mathematics Series, edited by, M. Abramowitz, and, I. Stegun, (U.S. Government Printing office, Washington, D.C., 1965), Vol. 55.
66. D. S. Jones, Elementary Information Theory (Oxford University Press, Oxford, 1979).
67. + δ).
68. The reader could object that this section is unnecessary, since the axiomatic derivation of the uncertainty function (which we call entropy) assumes that the entropy increases with the size of the system. However, the explicit statement of this axiom corresponds to the special case of uniform probabilities. Specifically, the axiom states that, if we have two systems A, B such that A contains n states a 1,..., a n and B contains n + 1 states, b 1,..., b n + 1, then, if ( i ≤ n) p (a i) = 1 / n and ( i ≤ n + 1) p (b i) = 1 / (n + 1) H (A) < H (B). Thus, if we are not dealing with this special case, we need to explicitly demonstrate that it holds for our purposes.
69. This derivation is equivalent to the one found in, Theorem 8.2. In this theorem, the authors demonstrate that every infinite distribution with infinite entropy is hyperbolic, which implies that the distribution is not dominated by a power law with an exponent higher than 1.