A brief history of generative models for power law and lognormal distributions

Internet Mathematics

Volumn 1, Issue 2, 2004, Pages 226-251

  Mitzenmacher, Michael ^a  

^a HARVARD UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 79960601554     PISSN: 15427951     EISSN: 19449488     Source Type: Journal    
DOI: 10.1080/15427951.2004.10129088     Document Type: Article

References (88)

1. [Aiello et al. 00] W. Aiello, F. Chung, and L. Lu. "A Random Graph Model for Massive Graphs". In Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, pp. 171-180. New York: ACM Press, 2000.
2. [Aitchison and Brown 54] J. Aitchison and J. A. C. Brown. "On Criteria for Descriptions of Income Distribution". Metroeconomica 6 (1954), 88-98.
3. [Aitchison and Brown 57] J. Aitchison and J. A. C. Brown. The Lognormal Distribution. Cambridge, UK: Cambridge University Press, 1957.
4. [Allen et al. 01] A. P. Allen, B. Li, and E. L. Charnov. "Population Fluctuations, Power Laws and Mixtures of Lognormal Distributions". Ecology Letters 4 (2001), 1-3.
5. [Albert et al. 99] R. Albert, H. Jeong, and A.-L. Barabási. "Diameter of the World Wide Web". Nature 401 (1999), 130-131.
6. [Auerbach 1913] F. Auerbach. "Das Gesetz der Bevolkerungskonzentration". Petermanns Geographische Mitteilungen LIX (1913), 73-76.
7. [Barabási et al. 99] A.-L. Barabási, R. Albert, and H. Jeong. "Mean-Field Theory for Scale-Free Random Networks". Physica A 272 (1999), 173-189.
8. [Barford et al. 99] P. Barford, A. Bestavros, A. Bradley, and M. Crovella. "Changes in Web Client Access Patterns: Characteristics and Caching Implications". World Wide Web 2 (1999), 15-28.
9. [Barford and Crovella 98] P. Barford and M. Crovella. "Generating Representative Web Workloads for Network and Server Performance Evaluation". In Proceedings of ACM SIGMETRICS, pp. 151-160. New York: ACM Press, 1998.
10. [Bell et al. 90] T. C. Bell, J. G. Cleary, and I. H. Witten. Text Compression. Englewood Cliffs, NJ: Prentice-Hall, 1990.
11. [Berry 99] B. Berry. "Déjà vu, Mr. Krugman". Urban Geography 20 (1999) 1-2.
12. [Black and Scholes 73] F. Black and M. Scholes. "The Pricing of Options and Corporate Liabilities". Journal of Political Economics 81 (1973), 637-654.
13. [Bollobás and Riordan to appear] B. Bollobás and O. Riordan. "The Diameter of a Scale-Free Random Graph". To appear.
14. [B. Bollobás et al. 01] B. Bollobás, O. Riordan, J. Spencer, and G. Tusnády. "The Degree Sequence of a Scale-Free Random Process". Random Structures and Algorithms 18:3 (2001), 279-290.
15. [Bookstein 90] A. Bookstein. "Informetric Distributions, Part I: Unified Overview". Journal of the American Society for Information Science 41:5 (1990), 368-375.
16. [Broder et al. 00] A. Broder, R. Kumar, F. Maghoul, P. Raghavan, S. Rajagopalan, R. Stata, A. Tomkins, and J. Wiener. "Graph Structure in the Web: Experiments and Models". In Proc. of the 9th World Wide Web Conference, pp. 309-320. Amsterdam: Elsevier Science, 2000.
17. [Carlson and Doyle 99] J. M. Carlson and J. Doyle. "Highly Optimized Tolerance: A Mechanism for Power Laws in Designed Systems". Physics Review E 60:2 (1999), 1412-1427.
18. [Champernowne 53] D. Champernowne. "A Model of Income Distribution". Economic Journal 63 (1953), 318-351.
19. [Cooper and Frieze 01] C. Cooper and A. Frieze. "On a General Model of Undirected Web Graphs". In Proceedings of the 9th Annual European Symposium on Algorithms, pp. 500-511. Berlin-Heidelberg: Springer-Verlag, 2001.
20. [Cox et al. 79] J. Cox, S. Ross, and M. Rubinstien. "Option Pricing: A Simplified Approach". Journal of Financial Economics 7 (1979), 229-265.
21. [Crovella and Bestavros 97] M. Crovella and A. Bestavros. "Self-Similarity in World Wide Web Traffic: Evidence and Possible Causes". IEEE/ACM Transactions on Networking 5:6 (1997), 835-846.
22. [Crovella et al. 98] M. Crovella, M. S. Taqqu, and A. Bestavros. "Heavy-Tailed Probability Distributions in the World Wide Web". In A Practical Guide to Heavy Tails, edited by R. J. Adler, R. E. Feldman, and M. S. Taqqu, pp. 3-26. London: Chapman and Hall, 1998.
23. [Crow and Shimizu 88] E. L. Crow and K. Shimizu (editors). Lognormal Distributions: Theory and Applications. New York: Markel Dekker, Inc., 1988.
24. [Drinea et al. 00] E. Drinea, M. Enachescu, and M. Mitzenmacher. "Variations on Random Graph Models of the Web". Harvard Computer Science Technical Report TR-06-01, 2000.
25. [Downey 01] A. B. Downey. "The Structural Causes of File Size Distributions". To appear in MASCOTS 2001. Available from World Wide Web (http://rocky.wellesley.edu/downey/filesize/), 2001.
26. [Estoup 1916] J. B. Estoup. Gammes Stenographiques. Paris: Institut Stenographique de France, 1916.
27. [Fabrikant et al. 02] A. Fabrikant, E. Koutsoupias, and C. H. Papadimitriou. "Heuristically Optimized Tradeoffs: A New Paradigm for Power Laws in the Internet". In Proceedings of the 29th International Colloquium on Automata, Languages, and Programming, Lecture Notes in Computer Science 2380, pp. 110-122. Berlin-Heidelberg: Springer-Verlag, 2002.
28. [Faloutsos et al. 99] M. Faloutsos, P. Faloutsos, and C. Faloutsos. "On Power-Law Relationships of the Internet Topology". In Proceedings of the ACM SIGCOMM 1999 Conference, pp. 251-261. New York: ACM Press, 1999.
29. [Gabaix 99] X. Gabaix. "Zipf's Law for Cities: An Explanation". Quarterly Journal of Economics 114 (1999), 739-767.
30. [Gibrat 30] R. Gibrat. "Une loi des réparations économiques: l'effet proportionnel". Bull. Statist. gén Fr. 19 (1930), 469.
31. [Gibrat 31] R. Gibrat. Les inegalites economiques. Paris: Libraire du Recueil Sirey, 1931.
32. [Gibson et al. 98] D. Gibson, J. Kleinberg, and P. Raghavan. "Inferring Web Communities from Link Topology". In Proceedings of the 9th ACM Conference on Hypertext and Hypermedia, pp. 225-234, New York: ACM Press, 1998.
33. [Gong et al. 01] W. Gong, Y. Liu, V. Misra, and D. Towsley. "On the Tails ofWeb Filesize Distributions". In Proceedings of the Thirty-Ninth Annual Allerton Conference on Communication, Control, and Computing, pp. 192-201. Urbana-Champaign, IL: Coordinated Science Laboratory, University of Illinois, 2001.
34. [Huberman and Adamic 99] B. A. Huberman and L. A. Adamic. "Evolutionary Dynamics of the World Wide Web". Technical Report, Xerox Palo Alto Research Center, 1999. Appears as a brief communication in Nature 401 (1999), 131.
35. [Huberman and Adamic 00] B. A. Huberman and L. A. Adamic. "The Nature of Markets in theWorldWideWeb". Quarterly Journal of Economic Commerce 1 (2000), 5-12.
36. [Hull 97] J. C. Hull. Introduction to Futures and Options Markets, Third edition. Engelwood Cliffs, NJ: Prentice-Hall, Inc., 1997.
37. [Ito 51] K. It̂o. Stochastic Differential Equations, Memoirs of the American Mathematical Society 4. Providence, RI: American Mathematical Society, 1951.
38. [Jain and Ramakumar 99] R. Jain and S. Ramakumar. "Stochastic Dynamics Modeling of the Protein Sequence Length Distribution in Genomes: Implications for Microbial Evolution". Physica A 273 (1999), 476-485.
39. [Kapteyn 1903] J. C. Kapteyn. Skew Frequency Curves in Biology and Statistics, Astronomical Laboratory. Groningen: Noordhoff, 1903.
40. [Kesten 73] H. Kesten. "Random Difference Equations and Renewal Theory for Products of Random Matrices". Acta Mathematica CXXXI (1973), 207-248.
41. [Kleinberg et al. 99] J. Kleinberg, R. Kumar, P. Raghavan, S. Rajagopalan, and A. Tomkins. "The Web as a Graph: Measurements, Models, and Methods". In Proceedings of the International Conference on Combinatorics and Computing, Lecture Notes in Computer Science 1627, pp. 1-18. Berlin-Heidelberg: Springer-Verlag, 1999.
42. [Krapivsky and Redner 01] P. L. Krapivsky and S. Redner. "Organization of Growing Random Networks". Physical Review E 63 (2001), 066123-1-066123-14.
43. [Krugman 96] P. Krugman. The Self-Organizing Economy. Cambridge, MA: Blackwell, 1996.
44. [Kumar et al. 99a] R. Kumar, P. Raghavan, S. Rajagopalan, and A. Tomkins. "Extracting Large Scale Knowledge Bases from the Web". In Proceedings of the 25th VLDB Conference, pp. 639-650. San Francisco: Morgan Kaufmann, 1999.
45. [Kumar et al. 99b] R. Kumar, P. Raghavan, S. Rajagopalan, and A. Tomkins. "Trawling the Web for Emerging Cyber-Communities". In Proceedings of the 8th International World Wide Web Conference, pp. 403-415. Amsterdam: Elsevier Science, 1999.
46. [Kumar et al 00] R. Kumar, P. Raghavan, S. Rajagopalan, D. Sivakumar, A. Tomkins, and E. Upfal. "Stochastic Models for the Web Graph". In Proceedings of the 41st Annual Symposium on Foundations of Computer Science, pp. 57-65. New York: ACM Press, 2000.
47. [Kurtz 81] T. G. Kurtz, Approximation of Population Processes. Philadelphia, PA: SIAM, 1981.
48. [Leland et al. 94] W. E. Leland, M. S. Taqqu, W. Willinger, and D. V. Wilson. "On the Self-Similar Nature of Ethernet Traffic". IEEE/ACM Transactions on Networking (1994), 1-15.
49. [Li 96] W. Li. "Random Texts Exhibit Zipf's-Law-Like Word Frequency Distribution". IEEE Transactions on Information Theory 38:6 (1992), 1842-1845.
50. [Li 99] W. Li. "References on Zipf's Law". Available from World Wide Web (http://linkage.rockerfeller.edu/wli/zipf/), 1999.
51. [Mihail and Papadimitriou 02] M. Mihail and C. H. Papadimitriou. "On the Eigenvalue Power Law". In Proceedings of RANDOM 2002, pp. 254-262. Berlin-Heidelberg: Springer-Verlag, 2002.
52. [Lotka 26] A. J. Lotka. "The Frequency Distribution of Scientific Productivity". Journal of the Washington Academy of Sciences 16 (1926), 317-323.
53. [Lu and Feng 98] J. Lu and Q. Feng. "Strong Consistency of the Number of the Vertices of Given Degrees in Nonuniform Random Recursive Trees". Yokohama Math. J. 45 (1998), 61-69.
54. [Mahmound et al. 93] H. Mahmound, R. Smythe, and J. Szymański. "On the Structure of Plane-Oriented Recursive Trees and their Branches. Random Structures and Algorithms 3 (1993), 255-266.
55. [Mandelbrot 53] B. Mandelbrot. "An Informational Theory of the Statistical Structure of Languages". In Communication Theory, edited by W. Jackson, pp. 486-502. Woburn, MA: Butterworth, 1953.
56. [Mandelbrot 59] B. Mandelbrot. "A Note on a Class of Skew Distribution Function: Analysis and Critique of a Paper by H.A. Simon". Information and Control 2 (1959), 90-99.
57. [Mandelbrot 61a] B. Mandelbrot. "Final Note on a Class of Skew Distribution Functions: Analysis and Critique of a Model due to H.A. Simon". Information and Control 4 (1961), 198-216.
58. [Mandelbrot 61b] B. Mandelbrot. "Post Scriptum to 'Final Note'". Information and Control 4 (1961), 300-304.
59. [Mandelbrot 97] B. Mandelbrot. Fractals and Scaling in Finance. New York: Springer-Verlag, 1997.
60. [McAlister 1879] D. McAlister. "The Law of the Geometric Mean". Proceedings of the Royal Society 29 (1879), 367.
61. [Medina et al. 00] A. Medina, I. Matta, and J. Byers. "On the Origin of Power Laws in Internet Topologies". Computer Communication Review 30:2 (2000), 18-28.
62. [Miller 57] G. A Miller. "Some Effects of Intermittent Silence". American Journal of Psychology 70 (1957), 311-314.
63. [Mitzenmacher 02] M. Mitzenmacher. "Dynamic Models for File Size Distributions and Double Pareto Distributions". Preprint, available from World Wide Web (http://www.eecs.harvard.edu/~michaelm/NEWWORK/papers/), 2002.
64. [Mitzenmacher 01] M. Mitzenmacher. "A Brief History of Generative Models for Power Law and Lognormal Distributions. In Proceedings of the Thirty-Ninth Annual Allerton Conference on Communication, Control, and Computing, pp. 182-191, Urbana-Champaign, IL: Coordinated Science Laboratory, University of Illinois, 2001.
65. [Montroll and Schlesinger 82] E. W. Montroll and M. F. Shlesinger. "On 1/f Noise and Other Distributions with Long Tails". Proceedings of the National Academy of Sciences 79 (1982), 3380-3383.
66. [Montroll and Schlesinger 83] E. W. Montroll and M. F. Shlesinger. "Maximum Entropy Formalism, Fractals, Scaling Phenomena, and 1/f Noise: A Tale of Tails". Journal of Statistical Physics 32 (1983), 209-230.
67. [Nakajima and Higurashi 98] T. Nakajima and A. Higurashi. "A Use of Two-Channel Radiances for an Aeresol Characterization from Space". Geophysical Research Letters 25 (1998), 3815-3818.
68. [Pareto 1896] V. Pareto. Cours d'Economie Politique. Geneva: Droz, 1896.
69. [Paxson and Floyd 95] V. Paxson and S. Floyd. "Wide-Area Traffic: The Failure of Poisson Modeling". IEEE/ACM Transactions on Networking 3 (1995), 226-244.
70. [Perline 96] R. Perline. "Zipf's Law, the Central Limit Theorem, and the Random Division of the Unit Interval". Physical Review E 54:1 (1996), 220-223.
71. [Pittel 94] B. Pittel. "Note on the Height of Recursive Trees and m-ary Search Trees". Random Structures and Algorithms 5 (1994), 337-347.
72. [Reed 00] W. J. Reed. "The Pareto Law of Incomes-An Explanation and an Extension". Submitted. Available from World Wide Web (http://www.math.uvic.ca/faculty/reed/index.html), 2000.
73. [Reed 01] W. J. Reed. "The Double Pareto-Lognormal Distribution-A New Parametric Model for Size Distribution". Available from World Wide Web (http://www.math.uvic.ca/faculty/reed/index.html), 2001.
74. [Reed and Hughes 02] W. J. Reed and B. D. Hughes. "From Gene Families and Genera to Incomes and Internet File Sizes: Why Power-Laws Are So Common in Nature". Available from World Wide Web (http://www.math.uvic.ca/faculty/reed/index.html), 2002.
75. [Simon 55] H. A. Simon. "On a Class of Skew Distribution Functions". Biometrika 42:3/4 (1955), 425-440.
76. [Simon 60] H. A. Simon. "Some Further Notes on a Class of Skew Distribution Functions". Information and Control 3 (1960), 80-88.
77. [Simon 61a] H. A. Simon. "Reply to 'Final Note'". Information and Control 4 (1961), 217-223.
78. [Simon 61b] H. A. Simon. "Reply to Dr. Mandelbrot's Post Scriptum". Information and Control 4 (1961), 305-308.
79. [Smythe and Mahmound 95] R. Smythe and H. Mahmound. "A Survey of Recursive Trees". Theoretical Probability and Mathematical Statistics 51 (1995) 1-27.
80. [Sole et al. 00] R. V. Sole, D. Alonso, and A. McKane. "Connectivity and Scaling in S-species Model Ecosystems". Physica A 286 (2000), 337-344.
81. [Szymański 87] J. Szymański. "On a Nonuniform Random Recursive Tree". Annals of Discrete Mathematics 333 (1987), 297-306.
82. [Wheatland and Sturrock 96] M. S. Wheatland and P. A. Sturrock. "Avalanche Models of Solar Flares and the Distribution of Active Regions". The Astrophysical Journal 471 (1996), 1044-1048.
83. [Wormald 95] N. C. Wormald. "Differential Equations for Random Processes and Random Graphs". Annals of Appl. Prob. 5 (1995), 1217-1235.
84. [Yule 25] G. Yule. "A Mathematical Theory of Evolution Based on the Conclusions of Dr. J.C. Willis". F.R.S. Philosophical Transactions of the Royal Society of London (Series B) 213 (1925), 21-87.
85. [Zhu et al. 01] X. Zhu, J. Yu, and J. Doyle. "Heavy Tails, Generalized Coding, and Optimal Web Layout". In Proceedings of IEEE INFOCOM, pp. 1617-1626. Los Alamitos, CA: IEEE Press, 2001.
86. [Zipf 32] G. Zipf. Selective Studies and the Principle of Relative Frequency in Language. Cambridge, MA: Harvard University Press, 1932.
87. [Zipf 35] G. Zipf. The Psycho-Biology of Language: An Introduction to Dynamic Philology. Boston, MA; Houghton Mifflin Company, 1935.
88. [Zipf 39] G. Zipf. Human Behavior and the Principle of Least Effort. Cambridge, MA: Addison-Wesley, 1949.