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 $r\_i$:for some fixed . We then want to minimize:$L\; =\; sum\_\{i=0\}^\{N-1\}\; p\_i\; x\_i$subject to the constraint:$sum\_\{i=0\}^\{N-1\}\; r\_i\; =\; kappa$Using Lagrange multipliers, this gives: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

* Carlson, J. M. & Doyle, J. (1999) "Phys. Rev. E" 60, 1412–1427.
* Carlson, J. M. & Doyle, J. (2000) "Phys. Rev. Lett." 84, 2529–2532.
* Doyle, J. & Carlson, J. M. (2000) "Phys. Rev. Lett." 84, 5656–5659.
* 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.
*
* 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.