Highly optimized tolerance

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. 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, [math]\displaystyle{ X }[/math], that takes on values [math]\displaystyle{ x_i }[/math] with probability [math]\displaystyle{ p_i }[/math]. Furthermore, let’s assume for another parameter [math]\displaystyle{ r_i }[/math]

[math]\displaystyle{ x_i = r_i^{ - \beta } }[/math]

for some fixed [math]\displaystyle{ \beta }[/math]. We then want to minimize

[math]\displaystyle{ L = \sum_{i=0}^{N-1} p_i x_i }[/math]

subject to the constraint

[math]\displaystyle{ \sum_{i=0}^{N-1} r_i = \kappa }[/math]

Using Lagrange multipliers, this gives

[math]\displaystyle{ p_i \propto x_i^{ - ( 1 + 1/ \beta) } }[/math]

giving us a power law. The global optimization of minimizing the energy along with the power law dependence between [math]\displaystyle{ x_i }[/math] and [math]\displaystyle{ r_i }[/math] gives us a power law distribution in probability.

See also

• self-organized criticality

References

• Carlson, J. M.; Doyle, John (August 1999), "Highly optimized tolerance: A mechanism for power laws in designed systems", Physical Review E 60 (2): 1412–1427, doi:10.1103/PhysRevE.60.1412, PMID 11969901, Bibcode: 1999PhRvE..60.1412C .
• Carlson, J. M.; Doyle, John (March 2000), "Highly Optimized Tolerance: Robustness and Design in Complex Systems", Physical Review Letters 84 (11): 2529–2532, doi:10.1103/PhysRevLett.84.2529, PMID 11018927, Bibcode: 2000PhRvL..84.2529C, https://authors.library.caltech.edu/1523/1/CARprl00.pdf .
• Doyle, John; Carlson, J. M. (June 2000), "Power Laws, Highly Optimized Tolerance, and Generalized Source Coding", Physical Review Letters 84 (24): 5656–5659, doi:10.1103/PhysRevLett.84.5656, PMID 10991018, Bibcode: 2000PhRvL..84.5656D, https://authors.library.caltech.edu/1524/1/DOYprl00.pdf .
• Greene, Katie (2005), "Untangling a web: The internet gets a new look", Science News 168 (15): 230, doi:10.2307/4016836, http://www.thefreelibrary.com/Untangling+a+Web%3A+the+Internet+gets+a+new+look.-a0138661490 .
• Li, Lun; Alderson, David; Doyle, John C.; Willinger, Walter (2005), "Towards a theory of scale-free graphs: definition, properties, and implications", Internet Mathematics 2 (4): 431–523, doi:10.1080/15427951.2005.10129111, http://projecteuclid.org/euclid.im/1150477667 .
• Robert, Carl; Carlson, J. M.; Doyle, John (April 2001), "Highly optimized tolerance in epidemic models incorporating local optimization and regrowth", Physical Review E 63 (5): 056122, doi:10.1103/PhysRevE.63.056122, PMID 11414976, Bibcode: 2001PhRvE..63e6122R, https://authors.library.caltech.edu/1526/1/ROBpre01.pdf .
• Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization and Disorder: Concepts and Tools, Springer Series in Synergetics, Berlin: Springer-Verlag, 2000, doi:10.1007/978-3-662-04174-1, ISBN 3-540-67462-4 .
• Zhou, Tong; Carlson, J. M. (2000), "Dynamics and changing environments in highly optimized tolerance", Physical Review E 62 (3): 3197–3204, doi:10.1103/PhysRevE.62.3197, PMID 11088814, Bibcode: 2000PhRvE..62.3197Z .
• Zhou, Tong; Carlson, J. M.; Doyle, John (2002), "Mutation, specialization, and hypersensitivity in highly optimized tolerance", Proceedings of the National Academy of Sciences 99 (4): 2049–2054, doi:10.1073/pnas.261714399, PMID 11842230, Bibcode: 2002PNAS...99.2049Z .