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.¹ 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.

The following is taken from Sornette's book.

Consider a random variable, ${\displaystyle X}$, that takes on values ${\displaystyle x_{i}}$ with probability ${\displaystyle p_{i}}$. Furthermore, let's assume for another parameter ${\displaystyle r_{i}}$

${\displaystyle x_{i}=r_{i}^{-\beta }}$

for some fixed ${\displaystyle \beta }$. We then want to minimize

${\displaystyle L=\sum _{i=0}^{N-1}p_{i}x_{i}}$

subject to the constraint

${\displaystyle \sum _{i=0}^{N-1}r_{i}=\kappa }$

Using Lagrange multipliers, this gives

${\displaystyle p_{i}\propto x_{i}^{-(1+1/\beta )}}$

giving us a power law. The global optimization of minimizing the energy along with the power law dependence between ${\displaystyle x_{i}}$ and ${\displaystyle r_{i}}$ gives us a power law distribution in probability.