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 and John Doyle 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 $$p_i$$. Furthermore, let’s 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.