ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Podlazov A. V. Competitors’ probability distribution law, state scale invariance and linear growth models. Izvestiya VUZ. Applied Nonlinear Dynamics, 2002, vol. 10, iss. 1, pp. 20-43. DOI: 10.18500/0869-6632-2002-10-1-20-43

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
(downloads: 0)
Article type: 

Competitors’ probability distribution law, state scale invariance and linear growth models

Podlazov Andrej Viktorovich, Keldysh Institute of Applied Mathematics (Russian Academy of Sciences)

The scale invariance is one of the distinguishers of the holistic behavior. Self-organized critical systems and systems with coherent noise giving examples of the scale invariance consist of essentially nonlinear elements. On the contrary, linear growth models with scale invariant properties are formulated in the paper.

Since the wholeness can’t arise from linear schemes, it is initially included in thе rules of the proposed models by means of using the information about the cumulative characteristics оf the system. The presence of such «seed оf the holistic properties» is quite enough for the distribution of system parts sizes to be power-law.

Besides that the questions of statistical samples processing for the construction of rank dependences, the analysis of the role of their parameters and the qualitative properties of systems described by power law probability distributions are considered in the paper.

Key words: 
This work was supported by RFBR (project No. 01-01-00628) and RGNF (project No. 99-03-19696).
  1. Podlazov AV. Self-organized criticality and risk analysis. Izvestiya VUZ. Applied Nonlinear Dynamics. 2001;9(1):49-88 (in Russian).
  2. Vladimirov VA, Vorobiev YL. Risk Management: Risk, Sustainable Development, Synergetics (Ser. Cybernetics: Unlimited Possibilities and Possible Limitations). Moscow: Nauka; 2000. 431 p. (in Russian).
  3. Khaitun SD. Scientometrics. Status and Prospects. Moscow: Nauka; 1983. 279 p. (in Russian).
  4. Khaitun SD. Problems of Quantitative Analysis of Science. Moscow: Nauka; 1989. 280 p. (in Russian).
  5. Petrov VM, Yablonsky AI. Mathematics and Social Processes: Hyperbolic Distributions and Their Applications. (Ser. Mathematics and Cybernetics). Moscow: Znanie; 1980. 64 p. (in Russian).
  6. Yablonsky AI. Mathematical Models in Science Research. Moscow: Nauka; 1986. 352 p. (in Russian).
  7. Zolotarev VM. One-Dimensional Stable Distributions (Ser. Probability Theory and Mathematical Statistics). Moscow: Nauka; 1983. 304 p. (in Russian).
  8. Zolotarev VM. Stable Laws and Their Applications (Ser. Mathematics and Cybernetics). No. 11. Moscow: Znanie; 1984. 64 p. (in Russian).
  9. Bak P. How Nature Works: The Science of Self-Organized Criticality. New York: Springer-Verlag; 1996. 212 p. DOI: 10.1007/978-1-4757-5426-1.
  10. Trubnikov BA. Law of distribution of competitors. Priroda. 1993;11:3-13 (in Russian).
  11. Piotrovsky RG, Bektaev KB, Piotrovskaya AA. Mathematical Linguistics. Moscow: Vysshaya Shkola; 1977. 383 p. (in Russian)
  12. Haggett P. Geography: A Modern Synthesis. New York: Harper & Row. James, P. E., and Jones; 1979. 627 p.
  13. Orlov YK. Invisible harmony. In: Number and Thought. Vol. 3. Moscow: Znanie; 1980. P. 70-106 (in Russian).
  14. Burlando В. The fractal dimension of taxonomic systems. J. Theor. Biol. 1990;146(1):99-114. DOI: 10.1016/S0022-5193(05)80046-3.
  15. Burlando В. The fractal geometry оf evolution. J. Theor. Biol. 1993;163(2):161-172. DOI: 10.1006/jtbi.1993.1114.
  16. Feller V. An Introduction to Probability Theory and Its Applications. 2nd ed. Vol. 2. Wiley; 1971. 704 p.
  17. Pivovarov YL. Fundamentals of Geourbanism. Urbanization and Urban Systems. Moscow: VLADOS; 1999. 232 p. (in Russian).
  18. Rank Countries by Population. Available from:
  19. Demographic Yearbook. Capital cities and cities оf 100,000 and more inhabitants. Available from:
  20. Newman MEJ, Sneppen K. Avalanches, scaling and coherent noise. Phys. Rev. Е. 1996;54(6):6226-6231. DOI: 10.1103/PhysRevE.54.6226.
  21. Newman MEJ. Self-organized criticality, evolution and the fossil extinction record. Proc. В. Soc. London В. 1996;263(1376):1605-1610. DOI: 10.1098/rspb.1996.0235.
  22. Sneppen K, Newman MEJ. Coherent noise, scale invariance and intermittency in large systems. Physica D. 1997;110(3-4):209-222. DOI: 10.1016/S0167-2789(97)00128-0.
  23. Malinetsky GG, Podlazov AV. The paradigm of self-organized criticality. Hierarchy of models and limits of predictability. Izvestiya VUZ. Applied Nonlinear Dynamics. 1997;5(5):89-106 (in Russian).
  24. Sornette D, Johansen А, Dornic I. Mapping self-organized criticality onto criticality. J. Phys. I (France). 1995;5(3):325-335. DOI: 10.1051/jp1:1995129.
  25. Podlazov AV. Model of surface release hecatonchiires and soft universality in the theory of self-organized criticality. Izvestiya VUZ. Applied Nonlinear Dynamics. 1999;7(6):3-16 (in Russian).
Available online: