# Highly optimized tolerance

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.

## Example

The following is taken from Sornette's book.

Consider a random variable, $X$, that takes on values $x_i$ with probability $p_i$. Furthmore, lets assume for another parameter $r_i$

$x_i = r_i^{ - \beta }$

for some fixed $\beta$. 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

$p_i \propto x_i^{ - ( 1 + 1/ \beta) }$

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.

## 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., 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.
• Sornette, Didier (2000). Critical Phenomena in Natural Sciences. Springer.
• 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.