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.

See also[edit]


  • Carlson, J. M. & Doyle, J. (1999) Phys. Rev. E 60, 1412–1427.
  • Carlson, J. M. & Doyle, J. (2000) Phys. Rev. Lett. 84, 2529–2532.
  • Doyle, J. & Carlson, J. M. (2000) Phys. Rev. Lett. 84, 5656–5659.
  • Greene, K. (2005) Science News 168, 230.
  • Li, L., Alderson, D., Tanaka, R., Doyle, J.C., Willinger, W., Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications (Extended Version). Internet Mathematics, 2005.
  • Robert, C., Carlson, J. M. & Doyle, J. (2001) Phys. Rev. E 63, 56122, 1–13.
  • Sornette, Didier (2000). Critical Phenomena in Natural Sciences. Springer. 
  • Zhou, T. & Carlson, J. M. (2000), Phys. Rev. E 62, 3197–3204.
  • Zhou, T., Carlson, J. M. & Doyle, J. (2002) Proc. Natl. Acad. Sci. USA 99, 2049–2054.