Highly optimized tolerance and power laws in dense and sparse resource regimes

Mary Elizabeth Manning, J. M. Carlson, J. Doyle

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

Power law cumulative frequency (P) versus event size (l) distributions P(≥l)∼l-α 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 α=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 α=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.

Original languageEnglish (US)
Article number016108
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Volume72
Issue number1
DOIs
StatePublished - Jul 2005
Externally publishedYes

Fingerprint

Tolerance
resources
Power Law
Resources
Loss Probability
Model
Exponent
Continuum Model
Continuous Distributions
exponents
Length Scale
Theoretical Model
Linking
Breakdown
Time Scales
Substrate
electric power
Model-based
files
Statistics

ASJC Scopus subject areas

  • Physics and Astronomy(all)
  • Condensed Matter Physics
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Highly optimized tolerance and power laws in dense and sparse resource regimes. / Manning, Mary Elizabeth; Carlson, J. M.; Doyle, J.

In: Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, Vol. 72, No. 1, 016108, 07.2005.

Research output: Contribution to journalArticle

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