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.

See also

• self-organized criticality

References

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