Scale-free behavior and universality in random fragmentation and aggregation

Jayanth R. Banavar; Paolo De Los Rios; Alessandro Flammini; Neal S. Holter; Amos Maritan

doi:10.1103/PhysRevE.69.036123

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Scale-free behavior and universality in random fragmentation and aggregation

Jayanth R. Banavar¹, Paolo De Los Rios², Alessandro Flammini^3,4, Neal S. Holter¹, and Amos Maritan^4,5

¹Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
²Laboratoire de Biophysique Statistique, ITP-SB, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
³International School for Advanced Studies, Trieste, Italy
⁴Unità INFM, Trieste and Abdus Salam International Center for Theoretical Physics, Trieste, Italy
⁵INFM and Dipartimento di Fisica "G. Galilei," Universita di Padova, Via Marzolo 8, 35131 Padova, Italy

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Phys. Rev. E 69, 036123 – Published 31 March, 2004

DOI: https://doi.org/10.1103/PhysRevE.69.036123

Abstract

Two distinct mechanisms underlying the existence of power-law distributions are presented: the distribution is stationary under the process of merging and splitting of classes and the distribution of the entities under study is invariant under changes of the classification scheme. We provide an explanation for the ubiquitous inverse n relationship in the species abundance relationship in ecology and the ${1 / n}^{2}$ distribution of company sizes based on the minimum impact principle.

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[c1] E.W. Montroll and M.F. Schlesinger, Proc. Natl. Acad. Sci. U.S.A. 79, 3380 (1982).

[c2] P. Bak, How Nature Works: The Science of Self-Organized Criticality (Springer-Verlag, New York, 1996).

[c3] H.E. Stanley, Rev. Mod. Phys. 71, S358 (1999).

[c4] R.N. Mantegna and H.E. Stanley, Introduction to Econophysics: Correlations and Complexity in Finance (Cambridge University Press, Cambridge, 2000).

[c5] A.L. Barabasi, Linked: The New Science of Networks (Perseus Press, Cambridge, MA, 2002).

[c6] M. Buchanan, Ubiquity: Why the World Is Simpler Than We Think (Weidenfeld & Nicholson, London, 2000).

[c7] M. Newman, http://www.santafe.edu/mark/powerlaws.pdf, 2002.

[c8] D. Sornette, Critical Phenomena in Natural Sciences (Springer, Berlin, 2000).

[c9] M. Smoluchowski, Phys. Z. 17, 557 (1916).

[c10] J.D. Barrow, J. Phys. A 14, 729 (1981).

[c11] P.L. Krapivsky, I. Grosse, and E. Ben-Naim, Phys. Rev. E 61, R993 (2000).

[c12] P.L. Krapivsky and S.N. Majumdar, Phys. Rev. Lett. 85, 5492 (2000).

[c13] K. Nagel, M. Shubik, M. Paczuski, and P. Bak, Physica A 287, 546 (2000).

[c14] G. Madras and B.J. McCoy, J. Colloid Interface Sci. 246, 356 (2002).

[c15] R.B. Diemer and J.H. Olson, Chem. Eng. Sci. 57, 2193 (2002).

[c16] D. Zheng, G.J. Rodgers, and P.M. Hui, Physica A 310, 480 (2002).

[c17] S. Gueron and S.A. Levin, Math. Biosci. 128, 243 (1995).

[c18] R.L. Axtell, Science 293, 1818 (2001).

[c19] R.M. May, in Ecology of Species and Communities, edited by M. Cody and J.M. Diamond (Harvard University Press, Cambridge, MA, 1975), pp. 81–120.

[c20] A.K. Dewdney, Math. Intell. 23, 27 (2001).

[c21] R.A. Fisher, A.S. Corbet, and C.B. Williams, J. Anim. Ecol. 12, 42 (1943).

[c22] B. Asfaw et al., Nature (London) 416, 317 (2002).

[c23] A.L. Roca et al., Science 293, 1473 (2001).

[c24] V. Novotny et al., Nature (London) 416, 841 (2002).

[c25] E. Mayr, Animal Species and Evolution (Harvard University Press, Cambridge, MA, 1963).

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