dorsal/arxiv
View SchemaHighly optimized tolerance and power laws in dense and sparse resource regimes
| Authors | M. Manning, J. M. Carlson, J. Doyle |
|---|---|
| Categories | |
| ArXiv ID | physics/0504136 |
| URL | https://arxiv.org/abs/physics/0504136 |
| DOI | 10.1103/PhysRevE.72.016108 |
Abstract
Power law cumulative frequency $(P)$ vs. event size $(l)$ distributions $P(\geq l)\sim l^{-\alpha}$ 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.
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"abstract": "Power law cumulative frequency $(P)$ vs. event size $(l)$ distributions\n$P(\\geq l)\\sim l^{-\\alpha}$ are frequently cited as evidence for complexity and\nserve as a starting point for linking theoretical models and mechanisms with\nobserved data. Systems exhibiting this behavior present fundamental\nmathematical challenges in probability and statistics. The broad span of length\nand time scales associated with heavy tailed processes often require special\nsensitivity to distinctions between discrete and continuous phenomena. A\ndiscrete Highly Optimized Tolerance (HOT) model, referred to as the\nProbability, Loss, Resource (PLR) model, gives the exponent $\\alpha=1/d$ as a\nfunction of the dimension $d$ of the underlying substrate in the sparse\nresource regime. This agrees well with data for wildfires, web file sizes, and\nelectric power outages. However, another HOT model, based on a continuous\n(dense) distribution of resources, predicts $\\alpha= 1+ 1/d $. In this paper we\ndescribe and analyze a third model, the cuts model, which exhibits both\nbehaviors but in different regimes. We use the cuts model to show all three\nmodels agree in the dense resource limit. In the sparse resource regime, the\ncontinuum model breaks down, but in this case, the cuts and PLR models are\ndescribed by the same exponent.",
"arxiv_id": "physics/0504136",
"authors": [
"M. Manning",
"J. M. Carlson",
"J. Doyle"
],
"categories": [
"physics.soc-ph"
],
"doi": "10.1103/PhysRevE.72.016108",
"title": "Highly optimized tolerance and power laws in dense and sparse resource regimes",
"url": "https://arxiv.org/abs/physics/0504136"
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