Highly optimized tolerance explained

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.

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 ^{- \beta} _{i} for some fixed \beta . We then want to minimizeL = \sum ^{N -1} _{i =0} p _{i} x _{i} subject to the constraint\sum ^{N -1} _{i =0} r _{i} = \kappa Using Lagrange multipliers, this givesp _{i} \propto x ^{- ( 1+1 / \beta)} _{i} 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

1. Carlson. null. Doyle. null. 2000-03-13. Highly optimized tolerance: robustness and design in complex systems. Physical Review Letters. 84. 11. 2529–2532. 10.1103/PhysRevLett.84.2529. 1079-7114. 11018927. 2000PhRvL..84.2529C.

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