Hisano Ryohei, Sornette Didier, Mizuno Takayuki
ETH Zurich, Department of Management, Technology and Economics, Kreuzplatz 5, CH-8092 Zurich, Switzerland.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 Aug;84(2 Pt 2):026117. doi: 10.1103/PhysRevE.84.026117. Epub 2011 Aug 24.
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.
齐普夫幂律分布是在许多复杂系统中发现的一种普遍的经验统计规律。然而，与具有单个幂律指数（齐普夫定律中等于1）的普遍性不同，有许多报告的偏差仍无法解释。最近发展的一种理论发现，（i）最普遍的因素之一，即随机比例增长，与（ii）出生和死亡过程之间的相互作用，导致了一种具有取决于每种因素特征的指数的普遍幂律分布。在此，我们基于对产品市场中市场份额动态的实证分析，报告了该理论及其应用的首次完整实证检验。我们直接估计市场份额的平均增长率及其标准差、产品的出生率和“死亡率”（风险率）。我们发现，观察到的幂律指数的时间变化和产品差异可以由该理论完全捕捉，且无需调整参数。我们的结果可以推广到许多系统，对于这些系统，幂律指数揭示的统计特性与潜在的生成机制直接相关。
