# 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. It was developed by Jean M. Carlson in the early 2000s. 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.