Power Laws, Highly Optimized Tolerance, and Generalized Source Coding

Abstract

We introduce a family of robust design problems for complex systems in uncertain environments which are based on tradeoffs between resource allocations and losses. Optimized solutions yield the "robust, yet fragile" features of highly optimized tolerance and exhibit power law tails in the distributions of events for all but the special case of Shannon coding for data compression. In addition to data compression, we construct specific solutions for world wide web traffic and forest fires, and obtain excellent agreement with measured data.

Optimal Design, Robustness, and Risk Aversion

“…2 provided too few data points to make an accurate evaluation of the exponent possible. We note also that the value − 3 2 for the two-dimensional case is quite different from the slope of − 1 2 measured for the cumulative size distribution of real forest fires [3,6]. Other functions with exponential tails also generate power laws, but give logarithmic corrections as well.…”

“…In a series of recent papers, Carlson and Doyle [1,2,3] have proposed a model for designed systems which they call "highly optimized tolerance" or HOT. The fundamental idea behind HOT is that systems designed for high performance naturally organize into highly structured, statistically unlikely states that are robust to perturbations they were designed to handle, yet fragile to rare perturbations and design flaws.…”

Wildfires, complexity, and highly optimized tolerance

“…The fact that fire statistics are indeed described by power laws was identified relatively recently (27), initially in the context of self-organized criticality (17). A more accurate statistical fit was later proposed by using HOT (22). A very recent analysis of fire size distributions has confirmed power law fits for many parts of the United States (28), but the mechanism generating these consistent patterns is still under investigation.…”

“…Theoretically, HOT builds on models and mathematics from physics and engineering, and identifies robustness tradeoffs as a principle underlying mechanism for complexity and power law statistics. HOT has been discussed in the context of a variety of technological and natural systems, including wildfires (18,22). A quantitative prediction for the distribution of fire sizes has come from an extremely simple analytical HOT model, referred to as the PLR (probability-loss-resource) model (22).…”

“…HOT has been discussed in the context of a variety of technological and natural systems, including wildfires (18,22). A quantitative prediction for the distribution of fire sizes has come from an extremely simple analytical HOT model, referred to as the PLR (probability-loss-resource) model (22). As a precursor to results presented later in this article, Fig.…”