Abstract.
Here we present a new model for Zipf's law in human word frequencies. The model defines the goal and the cost of communication using information theory. The model shows a continuous phase transition from a no communication to a perfect communication phase. Scaling consistent with Zipf's law is found in the boundary between phases. The exponents are consistent with minimizing the entropy of words. The model differs from a previous model [Ferrer i Cancho, Solé, Proc. Natl. Acad. Sci. USA 100, 788–791 (2003)] in two aspects. First, it assumes that the probability of experiencing a certain stimulus is controlled by the internal structure of the communication system rather than by the probability of experiencing it in the `outside' world, which makes it specially suitable for the speech of schizophrenics. Second, the exponent α predicted for the frequency versus rank distribution is in a range where α>1, which may explain that of some schizophrenics and some children, with α=1.5-1.6. Among the many models for Zipf's law, none explains Zipf's law for that particular range of exponents. In particular, two simplistic models fail to explain that particular range of exponents: intermittent silence and Simon's model. We support that Zipf's law in a communication system may maximize the information transfer under constraints.
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References
G.K. Zipf, Human behaviour and the principle of least effort. An introduction to human ecology (Hafner reprint, New York, 1972), 1st edn. (Cambridge, MA, Addison-Wesley, 1949)
V.K. Balasubrahmanyan S. Naranan, J. Quantitative Linguistics 3(3), 177 (1996)
R.G. Piotrowski, V.E. Pashkovskii, V.R. Piotrowski, Automatic Documentation and Mathematical Linguistics 28(5), 28 (1995), first published in Naučno-Techničeskaja Informatisija, Serija 2 28, No. 11, 21 (1994)
X. Piotrowska, W. Pashkovska, R. Piotrowski, to appear (2003)
L. Brilluen, Science and theory of information, Russian translation (Gos. Izd-vo Fiz,-Mat. Lit-ry, Moscow, 1960)
B. McCowan, L.R. Doyle, S.F. Hanser, J. Comparative Psychology 116, 166 (2002)
G.K. Zipf, Science 96, 344 (1942)
A.N. Kolguškin, Linguistic and engineering studies in automatic language translation of scientific Russian into English. Phase II (University of Washington Press, Seattle, 1970)
R. Suzuki, P.L. Tyack, J. Buck, Anim. Behav. (2003), accepted
G.A. Miller, Am. J. Psychol. 70, 311 (1957)
B. Mandelbrot, in Readings in mathematical social sciences, edited by P.F. Lazarsfield, N.W. Henry (MIT Press, Cambridge, 1966), pp. 151–168
G.A. Miller, N. Chomsky, in Handbook of Mathematical Psychology, edited by R.D. Luce, R. Bush, E. Galanter (Wiley, New York, 1963), Vol. 2
J.S. Nicolis, Chaos and information processing (World Scientific, Singapore, 1991)
W. Li, IEEE T. Inform. Theory 38(6), 1842 (November 1992)
H.A. Simon, Biometrika 42, 425 (1955)
S. Wolfram, A new kind of science (Wolfram Media, Champaign, 2002)
M.A. Nowak, J.B. Plotkin, V.A. Jansen, Nature 404, 495 (2000)
M.A. Nowak, J. Theor. Biol. 204, 179 (2000)
M.A. Nowak, Phil. Trans. R. Soc. Lond. B 355, 1615 (2000)
R. Suzuki, P.L. Tyack, J. Buck, Anim. Behav. 69, 9 (2005)
A. Cohen, R.N. Mantegna, S. Havlin, Fractals 5(1), 95 (1997)
R. Ferrer i Cancho, R.V. Solé, Adv. Complex Syst. 5, 1 (2002)
L.E. DeLisi, Schizophrenia Bulletin 27(3) (2001)
D.W. Carroll, Psychology of language (Brooks/Cole Publishing Company, Pacific Grove, California, 1994)
A. Akmajian, R.A. Demers, A.K. Farmer, R.M. Harnish, Linguistics. An Introduction to Language and Communication (MIT Press, 1995)
S. Manrubia, D. Zanette, J. Theor. Biol. 216, 461 (2002)
S. Franz, M. Leone, A. Montanari, F. Ricci-Tersenghi, Phys. Rev. E 66, 046120 (2002)
A. Montanari, N. Sourlas, Eur. Phys. J. B 18, 107 (2000)
A. Montanari, Eur. Phys. J. B 18, 121 (2000)
R. Ferrer i Cancho, R.V. Solé, Proc. Natl. Acad. Sci. USA 100, 788 (2003)
F. Pulvermüller, The neuroscience of language. On brain circuits of words and serial order (Cambridge University Press, Cambridge, 2003)
R. Ferrer i Cancho, O. Riordan, B. Bollobás, Proc. R. Soc. Lond. Series B 272, 561 (2005)
C.E. Shannon, Bell Systems Technical J. 27, 379 (1948)
Handbook of Psycholinguistics, edited by M.A. Gernsbacher (Academic Press, San Diego, 1994)
R.B. Ash, Information Theory (John Wiley & Sons, New York, 1965)
R. Ferrer i Cancho, Physica A 345, 275 (2004)
R. Ferrer i Cancho, Eur. Phys. J. B 44, 249 (2005)
N. Chomsky, Paper presented at the Univ. de Brasilia, Nov. 26 (1996)
C.F. Hockett, A course in modern linguistics (McMillan, New York, 1958)
K. von Frisch, Scientific American 207, 79 (1962)
K.T. Mueser, S.R. McGurk, The Lancet 363, 2063 (2004)
T.J. Crow, Schizophrenia Research 28, 127 (1997)
B. Elvevåg, T.E. Goldberg, Critical Reviews in Neurobiology 14, 1 (2000)
J. Binney, N. Dowrick, A. Fisher, M. Newman, The theory of critical phenomena. An introduction to the renormalization group (Oxford University Press, New York, 1992)
J. Tuldava, J. Quantitative Linguistics 3(1), 38 (1996)
R.J. Chitashvili, R.H. Baayen, in Quantitative Text Analysis, edited by G. Altmann, L. Hřebíček (Wissenschaftlicher Verlag Trier, Trier, 1993), pp. 54–135
A. Rapoport, Quantitative Linguistics 16, 1 (1982)
B. Mandelbrot, in Communication theory, edited by W. Jackson (Butterworths, London, 1953), p. 486
M.A. Montemurro, Physica A 300, 567 (2001)
A.A. Tsonis, C. Schultz, P.A. Tsonis, Complexity 3(5), 12 (1997)
I. Kanter, D.A. Kessler, Phys. Rev. Lett. 74, 4559 (1995)
S. Naranan, V.K. Balasubrahmanyan, Current Science 63, 261 (1992)
S. Naranan, V.K. Balasubrahmanyan, Current Science 63, 297 (1992)
S. Naranan, V.K. Balasubrahmanyan, J. Scientific and Industrial Research 52, 728 (1993)
P. Harremoës, F. Topsøe, Entropy 3, 227 (2001)
P. Harremoës, F. Topsøe, in Proceedings of the International Symposium on Information Theory, Lausanne, Switzerland (2002), p. 207
P. Allegrini, P. Gricolini, L. Palatella, Chaos, solitons and fractals 20, 95 (2004)
P. Allegrini, P. Gricolini, L. Palatella (World Scientific, 2003), submitted
A.G. Bashkirov, A.V. Vityazev, Physica A 277, 136 (2000)
A.G. Bashkirov, e-print: cond-mat/0211685 (2003)
T.J. Crow, British J. Psychiatry 173, 303 (1998)
M.A. Montemurro, D. Zanette, Glottometrics 4, 87 (2002)
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Ferrer i Cancho, R. Zipf's law from a communicative phase transition. Eur. Phys. J. B 47, 449–457 (2005). https://doi.org/10.1140/epjb/e2005-00340-y
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DOI: https://doi.org/10.1140/epjb/e2005-00340-y