Predicted and Verified Deviations from Zipf's law in Ecology of Competing Products

Zipf's power-law distribution is a generic empirical statistical regularity found in many complex systems. However, rather than universality with a single power-law exponent (equal to 1 for Zipf's law), there are many reported deviations that remain unexplained. A recently developed theory finds that the interplay between (i) one of the most universal ingredients, namely stochastic proportional growth, and (ii) birth and death processes, leads to a generic power-law distribution with an exponent that depends on the characteristics of each ingredient. Here, we report the first complete empirical test of the theory and its application, based on the empirical analysis of the dynamics of market shares in the product market. We estimate directly the average growth rate of market shares and its standard deviation, the birth rates and the "death" (hazard) rate of products. We find that temporal variations and product differences of the observed power-law exponents can be fully captured by the theory with no adjustable parameters. Our results can be generalized to many systems for which the statistical properties revealed by power law exponents are directly linked to the underlying generating mechanism

Statistical Properties of Business Firms Structure and Growth

We analyze a database comprising quarterly sales of 55624 pharmaceutical products commercialized by 3939 pharmaceutical firms in the period 1992--2001. We study the probability density function (PDF) of growth in firms and product sales and find that the width of the PDF of growth decays with the sales as a power law with exponent $\beta = 0.20 \pm 0.01$. We also find that the average sales of products scales with the firm sales as a power law with exponent $\alpha = 0.57 \pm 0.02$. And that the average number products of a firm scales with the firm sales as a power law with exponent $\gamma = 0.42 \pm 0.02$. We compare these findings with the predictions of models proposed till date on growth of business firms

Truncation of power law behavior in "scale-free" network models due to information filtering

We formulate a general model for the growth of scale-free networks under filtering information conditions--that is, when the nodes can process information about only a subset of the existing nodes in the network. We find that the distribution of the number of incoming links to a node follows a universal scaling form, i.e., that it decays as a power law with an exponential truncation controlled not only by the system size but also by a feature not previously considered, the subset of the network ``accessible'' to the node. We test our model with empirical data for the World Wide Web and find agreement.Comment: LaTeX2e and RevTeX4, 4 pages, 4 figures. Accepted for publication in Physical Review Letter

Power Law Scaling for a System of Interacting Units with Complex Internal Structure

We study the dynamics of a system composed of interacting units each with a complex internal structure comprising many subunits. We consider the case in which each subunit grows in a multiplicative manner. We propose a model for such systems in which the interaction among the units is treated in a mean field approximation and the interaction among subunits is nonlinear. To test the model, we identify a large data base spanning 20 years, and find that the model correctly predicts a variety of empirical results.Comment: 4 pages with 4 postscript figures (uses Revtex 3.1, Latex2e, multicol.sty, epsf.sty and rotate.sty). Submitted to PR

Pareto versus lognormal: a maximum entropy test

It is commonly found that distributions that seem to be lognormal over a broad range change to a power-law (Pareto) distribution for the last few percentiles. The distributions of many physical, natural, and social events (earthquake size, species abundance, income and wealth, as well as file, city, and firm sizes) display this structure. We present a test for the occurrence of power-law tails in statistical distributions based on maximum entropy. This methodology allows one to identify the true data-generating processes even in the case when it is neither lognormal nor Pareto. The maximum entropy approach is then compared with other widely used methods and applied to different levels of aggregation of complex systems. Our results provide support for the theory that distributions with lognormal body and Pareto tail can be generated as mixtures of lognormally distributed units

Mice lacking NF-κB1 exhibit marked DNA damage responses and more severe gastric pathology in response to intraperitoneal tamoxifen administration

Tamoxifen (TAM) has recently been shown to cause acute gastric atrophy and metaplasia in mice. We have previously demonstrated that the outcome of Helicobacter felis infection, which induces similar gastric lesions in mice, is altered by deletion of specific NF-κB subunits. Nfkb1-/- mice developed more severe gastric atrophy than wild-type (WT) mice 6 weeks after H. felis infection. In contrast, Nfkb2-/- mice were protected from this pathology. We therefore hypothesized that gastric lesions induced by TAM may be similarly regulated by signaling via NF-κB subunits. Groups of five female C57BL/6 (WT), Nfkb1-/-, Nfkb2-/- and c-Rel-/- mice were administered 150 mg/kg TAM by IP injection. Seventy-two hours later, gastric corpus tissues were taken for quantitative histological assessment. In addition, groups of six female WT and Nfkb1-/- mice were exposed to 12 Gy γ-irradiation. Gastric epithelial apoptosis was quantified 6 and 48 h after irradiation. TAM induced gastric epithelial lesions in all strains of mice, but this was more severe in Nfkb1-/- mice than in WT mice. Nfkb1-/- mice exhibited more severe parietal cell loss than WT mice, had increased gastric epithelial expression of Ki67 and had an exaggerated gastric epithelial DNA damage response as quantified by γH2AX. To investigate whether the difference in gastric epithelial DNA damage response of Nfkb1-/- mice was unique to TAM-induced DNA damage or a generic consequence of DNA damage, we also assessed gastric epithelial apoptosis following γ-irradiation. Six hours after γ-irradiation, gastric epithelial apoptosis was increased in the gastric corpus and antrum of Nfkb1-/- mice. NF-κB1-mediated signaling regulates the development of gastric mucosal pathology following TAM administration. This is associated with an exaggerated gastric epithelial DNA damage response. This aberrant response appears to reflect a more generic sensitization of the gastric mucosa of Nfkb1-/- mice to DNA damage

Size-dependent standard deviation for growth rates: empirical results and theoretical modeling

We study annual logarithmic growth rates R of various economic variables such as exports, imports, and foreign debt. For each of these variables we find that the distributions of R can be approximated by double exponential (Laplace) distributions in the central parts and power-law distributions in the tails. For each of these variables we further find a power-law dependence of the standard deviation σ(R) on the average size of the economic variable with a scaling exponent surprisingly close to that found for the gross domestic product (GDP) [Phys. Rev. Lett. 81, 3275 (1998)]. By analyzing annual logarithmic growth rates R of wages of 161 different occupations, we find a power-law dependence of the standard deviation σ(R) on the average value of the wages with a scaling exponent β≈0.14 close to those found for the growth of exports, imports, debt, and the growth of the GDP. In contrast to these findings, we observe for payroll data collected from 50 states of the USA that the standard deviation σ(R) of the annual logarithmic growth rate R increases monotonically with the average value of payroll. However, also in this case we observe a power-law dependence of σ(R) on the average payroll with a scaling exponent β≈−0.08. Based on these observations we propose a stochastic process for multiple cross-correlated variables where for each variable (i) the distribution of logarithmic growth rates decays exponentially in the central part, (ii) the distribution of the logarithmic growth rate decays algebraically in the far tails, and (iii) the standard deviation of the logarithmic growth rate depends algebraically on the average size of the stochastic variable

Polarization-analyzed small-angle neutron scattering. II. Mathematical angular analysis

Polarization-analyzed small-angle neutron scattering (SANS) is a powerful tool for the study of magnetic morphology with directional sensitivity. Building upon polarized scattering theory, this article presents simplified procedures for the reduction of longitudinally polarized SANS into terms of the three mutually orthogonal magnetic scattering contributions plus a structural contribution. Special emphasis is given to the treatment of anisotropic systems. The meaning and significance of scattering interferences between nuclear and magnetic scattering and between the scattering from magnetic moments projected onto distinct orthogonal axes are discussed in detail. Concise tables summarize the algorithms derived for the most commonly encountered conditions. These tables are designed to be used as a reference in the challenging task of extracting the full wealth of information available from polarization-analyzed SANS