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

Giant earthquakes and extreme wealth may not appear to have much in common, but the frequency with which the "Big One" will hit San Francisco and how often someone will earn as much money as Bill Gates can both be predicted with a statistical measurement called a power law exponent.

For the last century, researchers have used what's called a power law to predict certain kinds of events, including how frequently earthquakes at certain points on the Richter scale will occur. But a University of Michigan researcher noticed that this power law does not fit all circumstances.

Mitchell Newberry, a Michigan fellow and assistant professor at the U-M Center for the Study of Complex Systems, suggests an adjustment to the power law that would account for events that increase or decrease in fixed proportions—for example, when a manager makes roughly 20 percent more than his or her employee.

These adjustments affect how to estimate probabilities of earthquakes, the number of capillaries in the human body, and the sizes of megacities and solar flares. And they may revise when to expect the next Big One.

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

"The smoothness of this curve means that any value is possible," Newberry said. "I could make one penny more just as easily as one penny less."

That's not exactly the case with events such as earthquakes because of how they are recorded on the Richter scale. The Richter magnitude of earthquakes increases or decreases in increments of 0.1, exponentially. A magnitude 3.1 earthquake is 1.26 times as powerful as magnitude 3.0 earthquakes, so not every value is possible on the scale. The Richter scale is an example of a concept called "self-similarity," or when an event or thing is made of proportionately smaller copies of itself.

You can see self-similarity in nature as the branching of veins in a leaf, or in geometry as fitting triangles within larger triangles of the same shape, called a Sierpinski triangle. So, to account for events that change in exact proportions, 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 scientists are solving for. In earthquakes, that exponent, called the Gutenberg-Richter b value, was first measured in 1944 and indicates how often an earthquake of a certain strength is likely to occur. Newberry's discrete power law produced an 11.7% correction over estimates based on the continuous power law, bringing the exponent closer in line with the historical frequency of big earthquakes. Even a 5% correction translates to a more than twofold difference in when to expect the next giant .

"For 100 years, people have been talking about roughly one kind of power law distribution. It's the power law distribution of wealth and earthquakes," Newberry said. "Only now, we're documenting these discrete scales. Instead of a smooth curve, our power law looks like an infinite staircase."

Newberry noticed the flaw in the continuous power law in his study of the physics of the circulatory system. The circulatory system begins with one large blood vessel: the aorta. As the aorta splits into different branches—the carotid and subclavian arteries—each new branch decreases in diameter by roughly two-thirds.

He was using the continuous power law to estimate the sizes of blood vessels as they continue to branch. But the power law yielded sizes of blood vessels that could not occur. It indicated that a blood vessel might be only slightly smaller than the trunk from which it branched instead of around two-thirds of that trunk's size.

"Using the continuous power law, we were just getting answers we knew were wrong," Newberry said. "By debugging what failed, we figured out that this distribution makes the assumption that every blood vessel size is equally plausible. We know that for real vasculature, that's not the case."

So Newberry reverse-engineered the power law. By looking at blood vessels, Newberry could deduce the power law exponent from two constants: how many branches at each junction—two—and how much smaller each branch is relative to the trunk. Measuring vessel sizes at every division, Newberry was able to solve for the distribution of the blood vessels.

"There's some middle ground between a continuous power law and the discrete power law," Newberry said. "In the discrete power law, everything is laid out in perfectly rigid proportions from the highest scale to the infinitesimally small. In the continuous , everything is perfectly randomly laid out. Almost everything self-similar in reality is a mix of these two."

Newberry's study is published in the journal Physical Review Letters.

Explore further

Prediction of large earthquake probability improved

More 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
Journal information: Physical Review Letters

Feedback to editors

Apr 26, 2019
Great point, the differentiation of discrete from continuous! Were Vilfredo Pareto alive now, we could learn so much from him!

Benoit Mandelbrot said that "reality is fractally complex," and here is another demonstration.

Apr 27, 2019
"power law that predicts earthquakes" is a misleading sentence, because it does not allow to predict when the next earthquake will happen to us in our city, it remains totally random, unpredictable, only the statistic on thousend earthquakes over several K years is improved in this power law, taking into account of the statistically law of triggering the following earthquake, or next blood vessels branch, or next bank crash.

Apr 27, 2019
Their study increases by 10 fold the probability of a very big one earthquake over the predictions using extrapolations from the statistic of very small earthquakes ( very frequent below 3.5 ) using a straight line, in their figure 3 !!
Very frightning and scaring !!
read also in figure 2 quite more big one earthquakes or wildfires !!!!!!
https://arxiv.org...7868.pdf

Apr 27, 2019
A fundamental problem with "power laws", their derivation.
Generally, the values are not looked at directly, their logarithms are. The slope of the line produced is the "power". But logarithms can "squash" values so they fall near each other. Values that can differ from each other, based on the logarithm, can appear to be almost equal. And values that are scattered about can appear to be on a straight line.
Neutrons, dollar bills, snowflakes, shoes, couches, battleships and O5 type stars wear down by different mechanisms, but, if you use logarithms, the relationship between their mass and their lifetime, you can "conclude" that their mass controls their lifetime.

Apr 28, 2019
The log-periodic functions have been derived as a spin-off from Scale Relativity Theory and applied to evolutionary "trees" of all kinds already 20 years ago by Laurent Nottale, who published a ground breaking book at the time "https://www.amazo...355528", unfortunately still not translated to english today. Some of the specific applications can be read in dedicated articles as "https://www.luth....aix.pdf" and "https://www.luth....ash.pdf" also in french. More articles can be found here "https://www.luth....nlo.htm" and recently a book on SRT has been translated to english: https://www.amazo...38RYRJ/.
Stay wondered, Bernard

Apr 28, 2019
This seems a slam dunk model, as is the derivation from self similar processes. But notably in most cases there is not enough data, or care taken by the papers, to make a statistical test for a power distribution. When you check the papers on the popular (Mandelbrot, Pareto - as noted in Doug's comment) power law papers about half of those that can be tested are actually exponential distributions of simple growth.

@ dederau: They mean that it predicts the statistical distribution, obviously that cannot predict the individual stochastic event (if anything can, since it implies it is random in some sense).

Earthquakes less than 4 on most earthquake scales (there are several) are hardly noticeable, they are like a truck passing on the road [ https://en.wikipe...de_scale ]. They can be frequent, maybe since they may mostly be caused by thermal relaxation within a plate (fracturing) rather than a plate moving (as they are on planets without plate tectonics).

Apr 28, 2019
I don't understand why the links got corrupted:
The most relevant book: https://www.amazo...12355528
The most relevant articles in French: https://www.luth....Caix.pdf and https://www.luth....cash.pdf and more in English here
https://www.luth....wnlo.htm

Apr 28, 2019
While I composed my previous comment, this:

The log-periodic functions have been derived as a spin-off from Scale Relativity Theory and applied to evolutionary "trees" of all kinds already 20 years ago by Laurent Nottale,

That is exactly the type of eager pattern search that I argue we need less of!

It has of course been long recognized that a bifurcating phylogenetic tree is fractal in nature [ http://ib.berkele...tion.pdf ], that it a natural visualization [ https://www.ncbi....3472976/ ], and that fractal dimensions have meaning [ https://peerj.com.../198.pdf ].

- tbctd -

Apr 28, 2019
- ctd-

Nottale is an astrophysicist that neither Wikipedia (broken reference links) nor scientists makes much of [ https://en.wikipe..._Nottale ]. I can find one book chapter [ http://citeseerx....type=pdf ] where a triumvirate of one-topic authors [ https://www.resea...rre_Grou ] makes the lazy log-normal fit to an ad hoc assumption of fractal process spanning trees to human civilizations and then makes the opposite claim from fiat that fractal dimension is uninformative.

Garbage in, garbage out. And not very interesting for the average reader or scientist, as we can see.