Power-law distributions from additive preferential redistributions.

We introduce a nongrowth model that generates the power-law distribution with the Zipf exponent. There are elements, each of which is characterized by a quantity, and at each time step these quantities are redistributed through binary random interactions with a simple additive preferential rule, while the sum of quantities is conserved. The situation described by this model is similar to those of closed -particle systems when conservative two-body collisions are only allowed. We obtain stationary distributions of these quantities both analytically and numerically while varying parameters of the model, and find that the model exhibits the scaling behavior for some parameter ranges. Unlike well-known growth models, this alternative mechanism generates the power-law distribution when the growth is not expected and the dynamics of the system is based on interactions between elements. This model can be applied to some examples such as personal wealths, city sizes, and the generation of scale-free networks when only rewiring is allowed.

[1]  Lada A. Adamic,et al.  Internet: Growth dynamics of the World-Wide Web , 1999, Nature.

[2]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[3]  S. Redner,et al.  Wealth distributions in asset exchange models , 1998, 1006.4595.

[4]  Albert,et al.  Topology of evolving networks: local events and universality , 2000, Physical review letters.

[5]  S. N. Dorogovtsev,et al.  Structure of growing networks with preferential linking. , 2000, Physical review letters.

[6]  Hiroshi Yamamoto,et al.  Power-law tails in nonstationary stochastic processes with asymmetrically multiplicative interactions. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  M. Levy,et al.  POWER LAWS ARE LOGARITHMIC BOLTZMANN LAWS , 1996, adap-org/9607001.

[8]  M. E. J. Newman,et al.  Power laws, Pareto distributions and Zipf's law , 2005 .

[9]  D. Champernowne.  A Model of Income Distribution , 1953 .

[10]  H. Simon,et al.  ON A CLASS OF SKEW DISTRIBUTION FUNCTIONS , 1955 .

[11]  Michael Mitzenmacher,et al.  A Brief History of Generative Models for Power Law and Lognormal Distributions , 2004, Internet Math..

[12]  Albert-László Barabási,et al.  Internet: Diameter of the World-Wide Web , 1999, Nature.

[13]  A. Barabasi,et al.  Lethality and centrality in protein networks , 2001, Nature.

[14]  Arnab Chatterjee,et al.  Master equation for a kinetic model of a trading market and its analytic solution. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  S Redner,et al.  Degree distributions of growing networks. , 2001, Physical review letters.

[16]  Béla Bollobás,et al.  Robustness and Vulnerability of Scale-Free Random Graphs , 2004, Internet Math..

[17]  Victor M. Yakovenko,et al.  Statistical mechanics of money , 2000 .

[18]  George Kingsley Zipf,et al.  Human behavior and the principle of least effort , 1949 .

[19]  D. Zanette,et al.  ROLE OF INTERMITTENCY IN URBAN DEVELOPMENT : A MODEL OF LARGE-SCALE CITY FORMATION , 1997 .

[20]  X. Gabaix.  Zipf's Law for Cities: An Explanation , 1999 .

[21]  G. Yule,et al.  A Mathematical Theory of Evolution, Based on the Conclusions of Dr. J. C. Willis, F.R.S. , 1925 .

[22]  Power law distribution of wealth in population based on a modified Equíluz-Zimmermann model. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  Eli Ben-Naim,et al.  Self-similarity in random collision processes. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  Anders Hagson,et al.  Urban economy as a scale-free network. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  S. Redner.  How popular is your paper? An empirical study of the citation distribution , 1998, cond-mat/9804163.