Highly optimized tolerance and power laws in dense and sparse resource regimes.

Power law cumulative frequency (P) versus event size (l) distributions P > or =l) approximately l(-alpha) are frequently cited as evidence for complexity and serve as a starting point for linking theoretical models and mechanisms with observed data. Systems exhibiting this behavior present fundamental mathematical challenges in probability and statistics. The broad span of length and time scales associated with heavy tailed processes often require special sensitivity to distinctions between discrete and continuous phenomena. A discrete highly optimized tolerance (HOT) model, referred to as the probability, loss, resource (PLR) model, gives the exponent alpha=1/d as a function of the dimension d of the underlying substrate in the sparse resource regime. This agrees well with data for wildfires, web file sizes, and electric power outages. However, another HOT model, based on a continuous (dense) distribution of resources, predicts alpha=1+1/d . In this paper we describe and analyze a third model, the cuts model, which exhibits both behaviors but in different regimes. We use the cuts model to show all three models agree in the dense resource limit. In the sparse resource regime, the continuum model breaks down, but in this case, the cuts and PLR models are described by the same exponent.

[1]  Christos H. Papadimitriou,et al.  Heuristically Optimized Trade-Offs: A New Paradigm for Power Laws in the Internet , 2002, ICALP.

[2]  J M Carlson,et al.  Highly optimized tolerance: a mechanism for power laws in designed systems. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[3]  Michael Mitzenmacher,et al.  A Brief History of Generative Models for Power Law and Lognormal Distributions , 2004, Internet Math..

[4]  D. Turcotte,et al.  Forest fires: An example of self-organized critical behavior , 1998, Science.

[5]  Doyle,et al.  Highly optimized tolerance: robustness and design in complex systems , 2000, Physical review letters.

[6]  Walter Willinger,et al.  More "normal" than normal: scaling distributions and complex systems , 2004, Proceedings of the 2004 Winter Simulation Conference, 2004..

[7]  D. Applebaum.  Stable non-Gaussian random processes , 1995, The Mathematical Gazette.

[8]  Benoit B. Mandelbrot,et al.  Fractals and Scaling in Finance , 1997 .

[9]  David M. Raup,et al.  How Nature Works: The Science of Self-Organized Criticality , 1997 .

[10]  Jie Yu,et al.  Heavy tails, generalized coding, and optimal Web layout , 2001, Proceedings IEEE INFOCOM 2001. Conference on Computer Communications. Twentieth Annual Joint Conference of the IEEE Computer and Communications Society (Cat. No.01CH37213).

[11]  R. Gomory,et al.  Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Selecta Volume E , 1997 .

[12]  J. Doyle,et al.  Robust perfect adaptation in bacterial chemotaxis through integral feedback control. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[13]  J. Fowler,et al.  An Application of the Highly Optimized Tolerance Model to Electrical Blackouts , 2003, Int. J. Bifurc. Chaos.

[14]  Geoffrey J. McLachlan,et al.  Finite Mixture Models , 2019, Annual Review of Statistics and Its Application.

[15]  Michelle Girvan,et al.  Optimal design, robustness, and risk aversion. , 2002, Physical review letters.

[16]  Matthew Hennessy,et al.  Proceedings of the 29th International Colloquium on Automata, Languages and Programming , 2002 .

[17]  Claude E. Shannon,et al.  The mathematical theory of communication , 1950 .

[18]  W. Linde.  STABLE NON‐GAUSSIAN RANDOM PROCESSES: STOCHASTIC MODELS WITH INFINITE VARIANCE , 1996 .

[19]  P. Bak,et al.  A deterministic critical forest fire model , 1990 .

[20]  Drossel,et al.  Self-organized critical forest-fire model. , 1992, Physical review letters.

[21]  M. Taqqu,et al.  Stable Non-Gaussian Random Processes : Stochastic Models with Infinite Variance , 1995 .

[22]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[23]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[24]  Elmer S. West.  From the U. S. A. , 1965 .

[25]  Per Bak,et al.  How Nature Works , 1996 .

[26]  J. Doyle,et al.  Bow Ties, Metabolism and Disease , 2022 .

[27]  Doyle,et al.  Power laws, highly optimized tolerance, and generalized source coding , 2000, Physical review letters.

[28]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .

[29]  R. G. Ingalls,et al.  PROCEEDINGS OF THE 2002 WINTER SIMULATION CONFERENCE , 2002 .

[30]  宁北芳,et al.  疟原虫var基因转换速率变化导致抗原变异[英]／Paul H, Robert P, Christodoulou Z, et al//Proc Natl Acad Sci U S A , 2005 .

[31]  J. Stelling,et al.  Robustness of Cellular Functions , 2004, Cell.

[32]  L. Hood,et al.  Reverse Engineering of Biological Complexity , 2007 .

[33]  J M Carlson,et al.  Highly optimized tolerance in epidemic models incorporating local optimization and regrowth. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[35]  S. Redner,et al.  Introduction To Percolation Theory , 2018 .

[36]  John C. Doyle,et al.  Surviving heat shock: control strategies for robustness and performance. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[37]  J. Doyle,et al.  Design degrees of freedom and mechanisms for complexity. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[38]  John Doyle,et al.  Complexity and robustness , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[39]  Carlson,et al.  Dynamics and changing environments in highly optimized tolerance , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[40]  J. Doyle,et al.  Mutation, specialization, and hypersensitivity in highly optimized tolerance , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[41]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[42]  竹中 茂夫.  G.Samorodnitsky,M.S.Taqqu:Stable non-Gaussian Random Processes--Stochastic Models with Infinite Variance , 1996 .

[43]  Tang,et al.  Self-Organized Criticality: An Explanation of 1/f Noise , 2011 .