幂律粒度分布的不变性
The invariances of power law size distributions
摘要：
大小不等。小事情通常比大事情更频繁。频率的对数通常随尺寸的对数线性下降。这种幂律关系构成了自然界的一种普遍模式。为什么大自然的复杂性会简化成如此简单的模式？为什么像树的大小和酶的速率这样不同的事情都遵循类似的简单模式？在这里，我分析这些模式的不变性质。例如，当向所有观测添加常数值时，公共模式不应更改。这种移动本质上是对标尺上的点进行重新编号，而不改变标尺提供的度量信息。只有当标尺的刻度被正确地校准到被测量的图案时，标尺才是平移不变的。拉伸不变性对应于某物的总量守恒，例如总生物量和因此的平均大小。旋转不变性对应于不依赖于底层过程发生的顺序的模式，例如，将导致观察值的分量过程相加组合的尺度。我用树的大小作为一个例子来说明关键不变量是如何形成模式的。下面是对常见模式的简单解释。这种简单的解释通过基本不变性设定的尺度变换将正态分布与各种其他常见模式联系起来。
英文摘要：
Size varies. Small things are typically more frequent than large things. The logarithm of frequency often declines linearly with the logarithm of size. That power law relation forms one of the common patterns of nature. Why does the complexity of nature reduce to such a simple pattern? Why do things as different as tree size and enzyme rate follow similarly simple patterns? Here I analyze such patterns by their invariant properties. For example, a common pattern should not change when adding a constant value to all observations. That shift is essentially the renumbering of the points on a ruler without changing the metric information provided by the ruler. A ruler is shift invariant only when its scale is properly calibrated to the pattern being measured. Stretch invariance corresponds to the conservation of the total amount of something, such as the total biomass and consequently the average size. Rotational invariance corresponds to pattern that does not depend on the order in which underlying processes occur, for example, a scale that additively combines the component processes leading to observed values. I use tree size as an example to illustrate how the key invariances shape pattern. A simple interpretation of common pattern follows. That simple interpretation connects the normal distribution to a wide variety of other common patterns through the transformations of scale set by the fundamental invariances.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
