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Tuckahoe Joe’s Blog of the Week: Power Law Distributions.

January 20, 2019

 

Doug Jones (https://logarithmichistory.wordpress.com/) writes:

“On Boxing Day [December 26] 2004, a tsunami resulting from a 9.0+ magnitude earthquake killed about 250,000 people around the Indian Ocean. This was one of the deadliest natural disasters in recorded history. The Indian Ocean tsunami illustrated a major theme on this blog: the importance of catastrophe in human history, and in the history of life and the universe”

“Earthquakes are one example of a phenomenon following a power law statistical distribution. The frequency of earthquakes drops off as an exponential function of their magnitude so that on a logarithmic scale, the magnitude-frequency relationship looks linear. This is known as the Gutenberg-Ritter relation. (The deviation from linearity in the upper left part of the chart below may reflect measurement error, with a lot of tiny earthquakes not being detected.)”
pasted graphic
“Power law distributions are found in many other contexts, for example, in the frequency of wars versus their magnitude [as measured by the number of war deaths]. A power law distribution is very different from the more familiar bell-curve Gaussian normal distribution: extreme “black swan” events that are astronomically unlikely under a normal distribution may happen at an appreciable frequency under a power law distribution. Depending on the exponent, a power law distribution may not have a well-defined variance or even a well-defined mean.”

“For a technical discussion of why small scale processes sometimes aggregate to generate normally distributed outcomes, and other times aggregate to produce power-law distributions, here’s an article on The common patterns of nature. A take-home lesson — not always covered in introductory treatments of statistics and probability theory — is that catastrophes and extreme outcomes can be an expectable part of the natural order.”

“Finally, Steven Pinker and Nichlas Nassim Taleb have been squabbling about the implications of all this for the probability of a peaceful future. Here’s a level-headed review. And here are a couple of blog posts from me about why the bloody early twentieth century was maybe more than just a run of bad luck.”
https://logarithmichistory.wordpress.com/

By the way, the competing (or, a) theory is the famous and infamous “Bell Curve.” That placing the data points on a two vector grid events tend to congregate forming a hump or hill and if repeated, a wave. In other words, predicting the future of historical events on a two-axis graph produces either an inclined plane or a bell curve. Why this is so, I have no Idea. Maybe someday, I will find out. Right now, however, I couldn’t give a fig. (Actually, there is very little I would not give for a good fig.)

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