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# Researchers find a better power law that predicts earthquakes, blood vessels and bank accounts

Huge earthquakes and extreme prosperity do not seem to have much in common, but the frequency with which the "Big One" meets San Francisco, and how often someone earns as much money as Bill Gates, can both be predicted with a statistic measurement Power law exponent called.

In the last century, researchers have used a so-called power law to predict certain types of events, including the frequency with which earthquakes occur at certain points on the Richter scale. However, a University of Michigan researcher noted that this power law is not appropriate for all circumstances.

Mitchell Newberry, Michigan Fellow and Assistant Professor at UM Center for Study of Complex Systems, suggests adjusting the power law for events that increase or decrease in fixed proportions ̵

1; for example, when a manager is about 20 percent more than Employees earned.

These adjustments affect the estimation of the probability of earthquakes, the number of capillaries in the human body and the sizes of megacities and solar flares. And they can work out when to expect the next big one.

When scientists present something like the likelihood of extreme prosperity in a graph, the curve is a smooth line. That's because people can have any amount of money in their bank accounts.

"The smoothness of this curve means that every value is possible," Newberry said. "I could earn a penny more than a penny less."

This is not the case with events such as earthquakes, as they are recorded on the Richter scale. The Richter size of the earthquake increases or decreases exponentially in increments of 0.1. An earthquake of magnitude 3.1 is 1.26 times stronger than magnitude 3.0 earthquakes, so not every value on the scale is possible. The Richter scale is an example of a concept called "self-similarity" or when an event or thing is made from proportionally smaller copies of itself.

You can see self-similarity in nature as a branching of veins in a leaf, or in geometry as fitting triangles within larger triangles of the same shape, called the Sierpinski Triangle. To account for the events that are changing, Newberry and his co-author Van Savage of the University of California, Los Angeles, constructed the discrete power law.

In these power law equations, the exponent in the equation is the variable that scientists scrutinize. In earthquakes, this exponent, known as the Gutenberg Richter value b, was first measured in 1944 and indicates the number of times an earthquake of a certain magnitude is likely. The discrete power law of Newberry resulted in a correction of 11.7% over the estimates based on the continuous power law, bringing the exponent closer to the historical frequency of large earthquakes. Even a 5% correction results in a more than two-fold difference in when the next big earthquake is expected.

"For 100 years, earthquakes are being talked about about one type of power law distribution," Newberry said. "Only now are we documenting these discrete scales, instead of a smooth curve our power law looks like an infinite staircase."

Newberry noted the error of the continuous power law in his study of the physics of the circulatory system. The circulatory system starts with a large blood vessel: the aorta. While the aorta divides into several branches – the carotid artery and the subclavian artery – the diameter of each new branch decreases by about two-thirds.

He estimated the size of the blood vessels, while the aorta branched on, with the continuous power law. The law of power, however, revealed sizes of blood vessels that could not occur. It suggested that a blood vessel may be only slightly smaller than the trunk from which it branched, rather than about two thirds of the size of this strain.

"Through the Continuous Power Law, we received only answers that we knew were wrong." Said Newberry. "By debugging the failed error, we found that this distribution allows us to assume that every size of blood vessel is equally plausible, and we know that this is not the case for true vessels."

So Newberry has regressed the potency law. Using blood vessels, Newberry deduced the exponent of the power law from two constants: how many branches at each juncture – two – and how much smaller each branch is relative to the trunk. Newberry measured the vessel sizes in each department and was able to determine the distribution of the blood vessels.

"There is a middle ground between a continuous power law and the discrete power law," Newberry said. "In the discrete power law, everything is arranged in perfectly rigid ratios from the highest scale to the infinitely small scale, and in the continuous power law everything is randomly arranged." Almost everything that is similar in reality is a mixture of both

Newberrys The study is published in the journal Physical Review Letters .

Prediction of the probability of a major earthquake has been improved

Further information:
Mitchell G. Newberry et al. Self-similar processes follow a power law in discrete logarithmic space, Physical Review Letters (2019). DOI: 10.1103 / PhysRevLett.122.158303

Provided by
University of Michigan