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 internetlike graphs, forest fire models and may also apply to biological systems.
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 β. We then want to minimize
subject to the constraint
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.
