Zipf's law leads to heaps' law: Analyzing their relation in finite-size systems

PLoS ONE

Volumn 5, Issue 12, 2010, Pages

  Lü, Linyuan ^a,b   Zhang, Zi Ke ^b   Zhou, Tao ^a,b,c  

^a UNIVERSITY OF ELECTRONIC SCIENCE AND TECHNOLOGY OF CHINA   (China)

^b UNIVERSITY OF FRIBOURG   (Switzerland)

^c UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA   (China)

Author keywords

[No Author keywords available]

Indexed keywords

ARTICLE; CONTROLLED STUDY; FINITE SIZE SYSTEM; HEAPS LAW; MATHEMATICAL COMPUTING; NATURAL LANGUAGE PROCESSING; STATISTICAL ANALYSIS; STATISTICS; STOCHASTIC MODEL; ZIPF LAW; ALGORITHM; BIBLIOGRAPHIC DATABASE; BIOPHYSICS; GENETICS; LANGUAGE; METHODOLOGY; ORTHOMYXOVIRUS; PROBABILITY; STATISTICAL MODEL;

ALGORITHMS; BIOPHYSICS; DATABASES, BIBLIOGRAPHIC; LANGUAGE; MARKOV CHAINS; MODELS, STATISTICAL; ORTHOMYXOVIRIDAE; STOCHASTIC PROCESSES;

EID: 78649864468     PISSN: None     EISSN: 19326203     Source Type: Journal    
DOI: 10.1371/journal.pone.0014139     Document Type: Article

References (56)

1. Zipf GK (1949) Human Behaviour and the Principle of Least Effort: An introduction to human ecology (Addison-Wesly, Cambridge)
2. Heaps HS (1978) Information Retrieval: Computational and Theoretical Aspects (Academic Press, Orlando).
3. Clauset A, Shalizi CR, Newman MEJ (2009) Power-law distributions in empirical data. SIAM Rev 51: 661-703.
4. Axtell RL (2001) Zipf Distribution of U.S. Firm Sizes. Science 293: 1818-1820.
5. Dragulescu A, Yakovenko VM (2001) Exponential and power-law probability distributions of wealth and income in the United Kingdom and the United States. Physica A 299: 213-221.
6. Redner S (1998) How popular is your paper? An empirical study of the citation distribution. Eur Phys J B 4: 131-134.
7. Furusawa C, Kaneko K (2003) Zipf's Law in Gene Expression. Phys Rev Lett 90: 088102.
8. Bai W-J, Zhou T, Fu Z-Q, Chen Y-H, Wu X, Wang BH (2006) Electric power grids and blackouts in perspective of complex networks. Proc. 4th International Conference on Communications, Circuits and Systems (IEEE Press, New York). 2687 p.
9. Baek SK, Kiet HAT, Kim BJ (2007) Family name distributions: Master equation approach. Phys Rev E 76: 046113.
10. Cordoba JC (2008) On the distribution of city sizes. J Urban Econ 63: 177-197.
11. Chen Q, Wang C, Wang Y (2009) Deformed Zipf's law in personal donation. EPL 88: 38001.
12. Blasius B, Tönjes R (2009) Zipf's Law in the Popularity Distribution of Chess Openings. Phys Rev Lett 103: 218701.
13. Abhari A, Soraya M (2010) Workload generation for YouTube. Multimedia Tools Appl 46: 91-118.
14. Mitzenmacher M (2004) A brief history of generative models for power law and lognormal distributions. Internet Mathematics 1: 226-251.
15. Newman MEJ (2005) Power laws, Pareto distributions and Zipf's law. Contemporary Physics 46: 323-351.
16. Simon HA (1955) On a class of skew distribution functions. Biometrika 42: 425-440.
17. Barabasi A-L, Albert R (1999) Emergence of scaling in random networks. Science 286: 509-512.
18. Bak P (1996) How Nature Works: The Science of Self-Organized Criticality (Copernicus, New York).
19. Kanter I, Kessler DA (1995) Markov Processes: Linguistics and Zipf's Law. Phys Rev Lett 74: 4559-4562.
20. Marsili M, Zhang Y-C (1996) Interacting Individuals Leading to Zipf's Law. Phys Rev Lett 80: 2741-2744.
21. Carlson JM, Doyle J (1999) Highly optimized tolerance: A mechanism for power laws in designed systems. Phys Rev E 60: 1412-1427.
22. Cancho RFi, Solé RV (2003) Least effort and the origins of scaling in human language. Proc Natl Acad Sci USA 100: 788-791.
23. Ebeling W, Pöschel T (1994) Entropy and long range correlations in literary English. Europhys Lett 26: 241-246.
24. Kleinberg J (2003) Bursty and Hierarchical Structure in Streams. Data Min Knowl Disc 7: 373-397.
25. Altmann EG, Pierrehumbert JB, Motter AE (2009) Beyond Word Frequency: Bursts, Lulls, and Scaling in the Temporal Distributions of Words. PLoS ONE 4: e7678.
26. Gelbukh A, Sidorov G (2001) Zipf and Heaps Laws' Coefficients Depend on Language. Lect Notes Comput Sci. 2004: 332-335.
27. Serrano MA, Flammini A, Menczer F (2009) Modeling Statistical Properties of Written Text. PLoS ONE 4: e5372.
28. Cattuto C, Loreto V, Pietronero L (2007) Semiotic dynamics and collaborative tagging, Proc Natl Acad Sci USA 104: 1461-1464.
29. Cattuto C, Barrat A, Baldassarri A, Schehr G, Loreto V (2009) Collective dynamics of social annotation. Proc Natl Acad Sci USA 106: 10511-10515.
30. Zhang Z-K, L̈u L, Liu J-G, Zhou T (2008) Empirical analysis on a keyword-based semantic system. Eur Phys J B 66: 557-561.
31. Lansey JC, Bukiet B (2009) Internet Search Result Probabilities: Heaps' Law and Word Associativity. J Quant Linguistics 16: 40-66.
32. Zhang H-Y (2009) Discovering power laws in computer programs. Inf Process Manage 45: 477-483.
33. Benz RW, Swamidass SJ, Baldi P (2008) Discovery of Power-Laws in Chemical Space. J Chem Inf Model 48: 1138-1151.
34. Dorogovtsev SN, Mendes JFF (2001) Effect of the accelerating growth of communications networks on their structure. Phys Rev E 63: 025101.
35. Smith DMD, Onnela J-P, Johnson NF (2007) Accelerating networks. New J Phys 9: 181.
36. Broder A, Kumar R, Moghoul F, Raghavan P, Rajagopalan S, Stata R, Tomkins A, Wiener J (2000) Graph structure in the Web. Comput Netw 33: 309-320.
37. Zhang G-Q, Zhang G-Q, Yang Q-F, Cheng S-Q, Zhou T (2008) Evolution of the Internet and its cores. New J Phys 10: 123027.
38. Baeza-Yates RA, Navarro G (2000) Block addressing indices for approximate text retrieval. J Am Soc Inf Sci 51: 69-82.
39. van Leijenhorst DC, Weide Th P van der (2005) A formal derivation of Heaps' Law. Inf Sci 170: 263-272.
40. Mandelbrot B (1960) The pareto-levy law and the distribution of income. Int Econo Rev 1: 79-106.
41. Montemurro MA, Zanette DH (2002) New prespectives on Zipf's law in linguistics: from single texts to large corpora. Glottometrics 4: 87-99.
42. Zanette DH, Montemurro MA (2005) Dynamics of Text Generation with Realistic Zipf's Distribution. J Quant Linguistics 12: 29-40.
43. Goldstein ML, Morris SA, Yen GG (2004) Problems with fitting to the power-law distribution. Eur Phys J B 41: 255-258.
44. Carpena P, Bernaola-Galván P, Hackenberg M, Coronado AV, Oliver JL (2009) Level statistics of words: Finding keywords in literary texts and symbolic sequences. Phys Rev E 79: 035102.
45. Han XP, Wang BH, Zhou CS, Zhou T, Zhu JF (2009) Scaling in the Global Spreading Patterns of Pandemic Influenza A and the Role of Control: Empirical Statistics and Modeling, e-print arXiv. 0912.1390 p.
46. Albert R, Barabási A-L (2002) Statistical mechanics of complex networks. Rev Mod Phys 74: 47-97.
47. Newman MEJ (2003) The structure and function of complex networks. SIAM Rev 45: 167-256.
48. Ebel H, Mielsch L-I, Bornholdt S (2002) Scale-free topology of e-mail networks. Phys Rev E 66: 035103.
49. Liljeros F, Edling CR, Amaral LAN, Stanley HE, Åberg Y (2001) The web of human sexual contacts. Nature 411: 907-908.
50. Pastor-Satorras R, Vespignani A (2001) Epidemic Spreading in Scale-Free Networks. Phys Rev Lett 86: 3200-3203.
51. Vázquez A, Rácz B, Lukács A, Barabási A-L (2007) Impact of Non-Poissonian Activity Patterns on Spreading Processes. Phys Rev Lett 98: 158702.
52. Iribarren JL, Moro E (2009) Impact of Human Activity Patterns on the Dynamics of Information Diffusion. Phys Rev Lett 103: 038702.
53. Zhou T, Kiet HAT, Kim BJ, Wang BH, Holme P (2008) Role of activity in human dynamics. EPL 82: 28002.
54. Radicchi F (2009) Human activity in the web. Phys Rev E 80: 026118.
55. Caldarelli G (2007) Scale-Free Networks: Complex webs in nature and technology (Oxford University Press, Oxford). 197 p.
56. Corless RM, Gonnet GH, Hare DEG, Jeffrey DJ, Knuth DE (1996) On the Lambert W function. Adv Comput Math 5: 329-359.