Highly optimized tolerance - Alchetron, the free social encyclopedia

In applied mathematics, 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 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 . Furthmore, lets assume for another parameter r i

x i = r i − β

for some fixed β . We then want to minimize

L = ∑ i = 0 N − 1 p i x i

subject to the constraint

∑ i = 0 N − 1 r i = κ

Using Lagrange multipliers, this gives

p i ∝ x i − ( 1 + 1 / β )

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.

References

Highly optimized tolerance Wikipedia

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