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

Manning, M. and Carlson, J. M. and Doyle, J. (2005) Highly optimized tolerance and power laws in dense and sparse resource regimes. Physical Review E, 72 (1). Art. No. 016108. ISSN 1063-651X. http://resolver.caltech.edu/CaltechAUTHORS:MANpre05

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Abstract

Power law cumulative frequency (P) versus event size (l) distributions P(>= l)similar to 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.


Item Type:Article
Additional Information:©2005 The American Physical Society. Received 27 February 2005; published 8 July 2005. This work was supported by the David and Lucile Packard Foundation, NSF Grant No. DMR-9813752, the James S. McDonnell Foundation, and the Institute for Collaborative Biotechnologies through Grant No. DAAD19-03-D-0004 from the U.S. Army Research Office. M.M. was supported by the National Science Foundation.
Subject Keywords:probability; optimisation; stochastic processes
Record Number:CaltechAUTHORS:MANpre05
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:MANpre05
Alternative URL:http://dx.doi.org/10.1103/PhysRevE.72.016108
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:4185
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:07 Aug 2006
Last Modified:26 Dec 2012 08:58

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