论文标题
电力法的主方程
A Master Equation for Power Laws
论文作者
论文摘要
我们提出了一种新的机制来生成权力法律。从随机步行开始，我们首先概述了Fokker-Planck方程的简单推论。类比，从某个马尔可夫链开始，我们得出了一个幂律的主方程，该方程描述了级联反应的数量随时间变化（级联反向达到初始状态时结束的连续过渡）。部分微分方程具有封闭形式的解决方案，该解决方案具有明确的级联数量对其大小和按时的依赖性。此外，功率法解决方案具有自然截止，这是经验数据中经常看到的功能。这是由于级联在有限的时间范围内可以拥有的有限尺寸。方程式的推导为等于2的指数提供了理由，该指数与几个经验分布非常吻合，包括关于致命冲突规模和频率的法律。但是，可以为任何指数值求解该方程。此外，我们提出了一个urn模型，其中连续提取的数量遵循功率定律。在所有情况下，幂律都在整个级联大小范围内都表现出来，如频率和等级分布中的对数库图所示。
We propose a new mechanism for generating power laws. Starting from a random walk, we first outline a simple derivation of the Fokker-Planck equation. By analogy, starting from a certain Markov chain, we derive a master equation for power laws that describes how the number of cascades changes over time (cascades are consecutive transitions that end when the initial state is reached). The partial differential equation has a closed form solution which gives an explicit dependence of the number of cascades on their size and on time. Furthermore, the power law solution has a natural cut-off, a feature often seen in empirical data. This is due to the finite size a cascade can have in a finite time horizon. The derivation of the equation provides a justification for an exponent equal to 2, which agrees well with several empirical distributions, including Richardson's law on the size and frequency of deadly conflicts. Nevertheless, the equation can be solved for any exponent value. In addition, we propose an urn model where the number of consecutive ball extractions follows a power law. In all cases, the power law is manifest over the entire range of cascade sizes, as shown through log-log plots in the frequency and rank distributions.
