- Highly optimized tolerance
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 inernet-like graphs,
forest fire models and may also apply to biological systems.Example
The following is taken for Sornette's book.
Consider a
random variable , X, that takes on values x_i with probability p_i. Furthmore, lets assume for another parameter r_i:x_i = r_i^{ - eta }for some fixed eta. We then want to minimize:L = sum_{i=0}^{N-1} p_i x_i subject to the constraint:sum_{i=0}^{N-1} r_i = kappa UsingLagrange multipliers , this gives:p_i propto x_i^{ - ( 1 + 1/ eta) } giving us a power law. The global optimization of minimizing the energy along with the power law dependence between x_i and r_i gives us a power law distribution in probability.ee also
*
forest fires
*self-organized criticality
*power law References
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* Greene, K. (2005) "Science News" 168, 230.
* 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.
* Robert, C., Carlson, J. M. & Doyle, J. (2001) "Phys. Rev. E" 63, 56122, 1–13.
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* Zhou, T. & Carlson, J. M. (2000), "Phys. Rev. E" 62, 3197–3204.
* Zhou, T., Carlson, J. M. & Doyle, J. (2002) "Proc. Natl. Acad. Sci. USA" 99, 2049–2054.
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