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.{{Cite journal|last1=Carlson|first1=null|last2=Doyle|first2=null|date=2000-03-13|title=Highly optimized tolerance: robustness and design in complex systems|journal=Physical Review Letters|volume=84|issue=11|pages=2529–2532|doi=10.1103/PhysRevLett.84.2529|issn=1079-7114|pmid=11018927|url=https://authors.library.caltech.edu/1523/1/CARprl00.pdf|bibcode=2000PhRvL..84.2529C}} 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 $r_i$

: $x_i = r_i^{ - \beta }$

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.

References

{{citation

| last1 = Carlson | first1 = J. M.

| last2 = Doyle | first2 = John

| date = August 1999

| doi = 10.1103/PhysRevE.60.1412

| issue = 2

| journal = Physical Review E

| pages = 1412–1427

| title = Highly optimized tolerance: A mechanism for power laws in designed systems

| volume = 60| pmid = 11969901

| arxiv = cond-mat/9812127| bibcode = 1999PhRvE..60.1412C| s2cid = 2648280

| author1-link = Jean M. Carlson

}}.

{{citation

| last1 = Carlson | first1 = J. M.

| last2 = Doyle | first2 = John

| date = March 2000

| doi = 10.1103/PhysRevLett.84.2529

| issue = 11

| journal = Physical Review Letters

| pages = 2529–2532

| title = Highly Optimized Tolerance: Robustness and Design in Complex Systems

| volume = 84

| bibcode=2000PhRvL..84.2529C| url = https://authors.library.caltech.edu/1523/1/CARprl00.pdf

| pmid = 11018927

}}.

{{citation

| last1 = Doyle | first1 = John

| last2 = Carlson | first2 = J. M.

| date = June 2000

| doi = 10.1103/PhysRevLett.84.5656

| issue = 24

| journal = Physical Review Letters

| pages = 5656–5659

| title = Power Laws, Highly Optimized Tolerance, and Generalized Source Coding

| volume = 84

| bibcode=2000PhRvL..84.5656D

| pmid=10991018| url = https://authors.library.caltech.edu/1524/1/DOYprl00.pdf

}}.

{{citation

| last = Greene | first = Katie

| doi = 10.2307/4016836

| issue = 15

| journal = Science News

| pages = 230

| title = Untangling a web: The internet gets a new look

| url = http://www.thefreelibrary.com/Untangling+a+Web%3A+the+Internet+gets+a+new+look.-a0138661490

| volume = 168

| year = 2005| jstor = 4016836

}}.

{{citation

| last1 = Li | first1 = Lun

| last2 = Alderson | first2 = David

| last3 = Doyle | first3 = John C.

| last4 = Willinger | first4 = Walter

| arxiv = cond-mat/0501169

| issue = 4

| journal = Internet Mathematics

| mr = 2241756

| pages = 431–523

| title = Towards a theory of scale-free graphs: definition, properties, and implications

| url = http://projecteuclid.org/euclid.im/1150477667

| volume = 2

| year = 2005 | doi=10.1080/15427951.2005.10129111| s2cid = 107

}}.

{{citation

| last1 = Robert | first1 = Carl

| last2 = Carlson | first2 = J. M.

| last3 = Doyle | first3 = John

| date = April 2001

| doi = 10.1103/PhysRevE.63.056122

| issue = 5

| journal = Physical Review E

| page = 056122

| title = Highly optimized tolerance in epidemic models incorporating local optimization and regrowth

| volume = 63| pmid = 11414976

| bibcode = 2001PhRvE..63e6122R| url = https://authors.library.caltech.edu/1526/1/ROBpre01.pdf

}}.

{{citation

| last = Sornette | first = Didier | author-link = Didier Sornette

| doi = 10.1007/978-3-662-04174-1

| isbn = 3-540-67462-4

| location = Berlin

| mr = 1782504

| publisher = Springer-Verlag

| series = Springer Series in Synergetics

| title = Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization and Disorder: Concepts and Tools

| year = 2000}}.

{{citation

| last1 = Zhou | first1 = Tong

| last2 = Carlson | first2 = J. M.

| doi = 10.1103/PhysRevE.62.3197

| journal = Physical Review E

| pages = 3197–3204

| title = Dynamics and changing environments in highly optimized tolerance

| volume = 62

| issue = 3

| year = 2000| pmid = 11088814

| bibcode = 2000PhRvE..62.3197Z}}.

{{citation

| last1 = Zhou | first1 = Tong

| last2 = Carlson | first2 = J. M.

| last3 = Doyle | first3 = John

| doi = 10.1073/pnas.261714399

| issue = 4

| journal = Proceedings of the National Academy of Sciences

| pages = 2049–2054

| title = Mutation, specialization, and hypersensitivity in highly optimized tolerance

| volume = 99

}}.

Category:Mathematical optimization

Highly optimized tolerance

Example

See also

References