Highly optimized tolerance

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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.


The following is taken from Sornette's book.

Consider a random variable, , that takes on values with probability . Furthmore, lets assume for another parameter

for some fixed . 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 and gives us a power law distribution in probability.

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