Highly optimized tolerance

The following is taken from Sornette's book.

Consider a random variable, ${\displaystyle X}$, that takes on values ${\displaystyle x_{i}}$ with probability ${\displaystyle p_{i}}$. Furthermore, let’s assume for another parameter ${\displaystyle r_{i}}$

${\displaystyle x_{i}=r_{i}^{-\beta }}$

for some fixed ${\displaystyle \beta }$. We then want to minimize

${\displaystyle L=\sum _{i=0}^{N-1}p_{i}x_{i}}$

subject to the constraint

${\displaystyle \sum _{i=0}^{N-1}r_{i}=\kappa }$

Using Lagrange multipliers, this gives

${\displaystyle p_{i}\propto x_{i}^{-(1+1/\beta )}}$

giving us a power law. The global optimization of minimizing the energy along with the power law dependence between ${\displaystyle x_{i}}$ and ${\displaystyle r_{i}}$ gives us a power law distribution in probability.

• self-organized criticality

• 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, arXiv:cond-mat/9812127, Bibcode:1999PhRvE..60.1412C, doi:10.1103/PhysRevE.60.1412, PMID 11969901, S2CID 2648280.
• Carlson, J. M.; Doyle, John (March 2000), "Highly Optimized Tolerance: Robustness and Design in Complex Systems" (PDF), Physical Review Letters, 84 (11): 2529–2532, Bibcode:2000PhRvL..84.2529C, doi:10.1103/PhysRevLett.84.2529, PMID 11018927.
• Doyle, John; Carlson, J. M. (June 2000), "Power Laws, Highly Optimized Tolerance, and Generalized Source Coding" (PDF), Physical Review Letters, 84 (24): 5656–5659, Bibcode:2000PhRvL..84.5656D, doi:10.1103/PhysRevLett.84.5656, PMID 10991018.
• Greene, Katie (2005), "Untangling a web: The internet gets a new look", Science News, 168 (15): 230, doi:10.2307/4016836, JSTOR 4016836.
• 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, arXiv:cond-mat/0501169, doi:10.1080/15427951.2005.10129111, MR 2241756, S2CID 107.
• Robert, Carl; Carlson, J. M.; Doyle, John (April 2001), "Highly optimized tolerance in epidemic models incorporating local optimization and regrowth" (PDF), Physical Review E, 63 (5): 056122, Bibcode:2001PhRvE..63e6122R, doi:10.1103/PhysRevE.63.056122, PMID 11414976.
• Sornette, Didier (2000), Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization and Disorder: Concepts and Tools, Springer Series in Synergetics, Berlin: Springer-Verlag, doi:10.1007/978-3-662-04174-1, ISBN 3-540-67462-4, MR 1782504.
• Zhou, Tong; Carlson, J. M. (2000), "Dynamics and changing environments in highly optimized tolerance", Physical Review E, 62 (3): 3197–3204, Bibcode:2000PhRvE..62.3197Z, doi:10.1103/PhysRevE.62.3197, PMID 11088814.
• 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, Bibcode:2002PNAS...99.2049Z, doi:10.1073/pnas.261714399, PMC 122317, PMID 11842230.