Manning M, Carlson J M, Doyle J
Department of Physics, University of California, Santa Barbara, California 93106, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2005 Jul;72(1 Pt 2):016108. doi: 10.1103/PhysRevE.72.016108. Epub 2005 Jul 8.
Power law cumulative frequency (P) versus event size (l) distributions P > or =l) approximately l(-alpha) are frequently cited as evidence for complexity and serve as a starting point for linking theoretical models and mechanisms with observed data. Systems exhibiting this behavior present fundamental mathematical challenges in probability and statistics. The broad span of length and time scales associated with heavy tailed processes often require special sensitivity to distinctions between discrete and continuous phenomena. A discrete highly optimized tolerance (HOT) model, referred to as the probability, loss, resource (PLR) model, gives the exponent alpha=1/d as a function of the dimension d of the underlying substrate in the sparse resource regime. This agrees well with data for wildfires, web file sizes, and electric power outages. However, another HOT model, based on a continuous (dense) distribution of resources, predicts alpha=1+1/d . In this paper we describe and analyze a third model, the cuts model, which exhibits both behaviors but in different regimes. We use the cuts model to show all three models agree in the dense resource limit. In the sparse resource regime, the continuum model breaks down, but in this case, the cuts and PLR models are described by the same exponent.
幂律累积频率（P）与事件规模（l）的分布P≥l）近似为l^(-α)，常被引为复杂性的证据，并作为将理论模型和机制与观测数据相联系的起点。表现出这种行为的系统在概率和统计方面存在基本的数学挑战。与重尾过程相关的长度和时间尺度范围广泛，常常需要对离散和连续现象之间的区别有特殊的敏感性。一种离散的高度优化容限（HOT）模型，称为概率、损失、资源（PLR）模型，在稀疏资源 regime 中给出指数α = 1/d，作为基础底物维度d的函数。这与野火、网页文件大小和电力中断的数据非常吻合。然而，另一种基于资源连续（密集）分布的HOT模型预测α = 1 + 1/d。在本文中，我们描述并分析了第三种模型，即切割模型，它在不同 regime 中表现出这两种行为。我们使用切割模型表明，所有三种模型在密集资源极限下是一致的。在稀疏资源 regime 中，连续统模型失效，但在这种情况下，切割模型和PLR模型由相同的指数描述。
