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| | | Jayson Berryhill is how I'm known as and my spouse doesn't like it at all. Ohio is where his house is and his family members loves it. I am currently a journey agent. My husband doesn't like it the way I do but what I really like performing is caving but I don't have the time lately.<br><br>Here is my web-site - live psychic reading - [http://kjhkkb.net/xe/notice/374835 http://kjhkkb.net/xe/notice/374835], |
| {{technical|date=June 2012}}
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| In [[applied mathematics]], '''highly optimized tolerance (HOT)''' is a method of generating [[power law]] behavior in systems by including a [[global optimization]] principle. For some systems that display a characteristic scale, a global optimization term could potentially be added that would then yield power law behavior. It has been used to generate and describe internet-like graphs, [[forest fire]] models and may also apply to biological systems.
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| ==Example==
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| The following is taken from Sornette's book.
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| Consider a [[random variable]], <math>X</math>, that takes on values <math>x_i</math> with probability <math>p_i</math>. Furthmore, lets assume for another parameter <math>r_i</math>
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| :<math>x_i = r_i^{ - \beta }</math>
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| for some fixed <math>\beta</math>. We then want to minimize
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| :<math> L = \sum_{i=0}^{N-1} p_i x_i </math>
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| subject to the constraint
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| :<math> \sum_{i=0}^{N-1} r_i = \kappa </math>
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| Using [[Lagrange multipliers]], this gives
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| :<math> p_i \propto x_i^{ - ( 1 + 1/ \beta) } </math> | |
| giving us a power law. The global optimization of minimizing the energy along with the power law dependence between <math>x_i</math> and <math>r_i</math> gives us a power law distribution in probability.
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| ==See also==
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| * [[self-organized criticality]]
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| ==References==
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| * Carlson, J. M. & Doyle, J. (1999) ''Phys. Rev. E'' '''60,''' 1412–1427.
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| * Carlson, J. M. & Doyle, J. (2000) ''Phys. Rev. Lett.'' '''84,''' 2529–2532.
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| * Doyle, J. & Carlson, J. M. (2000) ''Phys. Rev. Lett.'' '''84,''' 5656–5659.
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| * Greene, K. (2005) ''Science News'' '''168,''' 230.
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| * Li, L., Alderson, D., Tanaka, R., Doyle, J.C., Willinger, W., [http://arxiv.org/abs/cond-mat/0501169 Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications (Extended Version)]. Internet Mathematics, 2005.
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| * Robert, C., Carlson, J. M. & Doyle, J. (2001) ''Phys. Rev. E'' '''63''', 56122, 1–13.
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| * {{cite book|last=Sornette|first=Didier|authorlink=Didier Sornette|title=Critical Phenomena in Natural Sciences|year=2000|publisher=Springer}}
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| * Zhou, T. & Carlson, J. M. (2000), ''Phys. Rev. E'' '''62,''' 3197–3204.
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| * Zhou, T., Carlson, J. M. & Doyle, J. (2002) ''Proc. Natl. Acad. Sci. USA'' '''99,''' 2049–2054.
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| [[Category:Mathematical optimization]]
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| {{sci-stub}}
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