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, X, that takes on values x_i with probability p_i. Furthmore, lets 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.

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