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, <math>X</math>, that takes on values <math>x_i</math> with probability <math>p_i</math>. Furthermore, let's assume for another parameter <math>r_i</math>
:<math>x_i = r_i^{ - \beta }</math>
for some fixed <math>\beta</math>. We then want to minimize
:<math> L = \sum_{i=0}^{N-1} p_i x_i </math>
subject to the constraint
:<math> \sum_{i=0}^{N-1} r_i = \kappa </math>
Using Lagrange multipliers, this gives
:<math> p_i \propto x_i^{ - ( 1 + 1/ \beta) } </math>
giving us a power law. The global optimization of minimizing the energy along with the power law dependence between <math>x_i</math> and <math>r_i</math> gives us a power law distribution in probability.
See also
- self-organized criticality