On Power Laws
06 January, 2022 - 7 min read
Power law is one form of Pareto's principle where the proportion of 80/20 shifts to extreme proportion.
Power law is a well known concept in seismology, the study of earthquakes. A large scale earthquake releases twice the energy than a small scale earthquake. A large scale earthquake can cause a lot of damage, but they occur rarely. In California, small scale earthquakes are common, but the damage is negligible. However, the impact of one big earthquake can be bigger than the sum of millions of smaller and more common ones. This is one example of power laws.
Another example of power law is wealth distribution. Often you hear the top 1% of American household owns 99% of the wealth. The combined wealth of top 1% is orders of magnitude greater than all of 99% combined. This pattern of income distribution can be attributed to the mathematical principle of power law. This is why billionaires are a tiny fraction of our total population.
A relationship between two quantities whereby a small change in one, results in a large change in the other. If you double the diameter of a circle, the area would quadruple. Power law is an extension of Pareto's principle which indicates 80% of the outcomes comes from 20% of the action. For example, 80% of the sales come from 20% of the customers; 80% of the world's internet traffic go to 20% of the websites. But a power law skews towards a more extreme proportion. For example, 99% of the traffic goes to 1% of the websites.
There is a rich and long history of power law distribution spanning in many fields. Power law distributions have many names, often referred to as long-tail distributions, Pareto distributions, Zipfian distributions, Benford's law, Stefan-Boltzmann law and Steven's power law. The idea of a power law had been suggested by 19th-century researchers, but much of the recent interest in power laws has come recently, specifically from the study of probability distributions.
Power law equation
It is primarily in the study of statistical distributions that the name “power law” is used, but mathematically, a power law cannot be a probability distribution, but a distribution that is a power function involving two numbers, the base and the exponent.
In math class, we all once asked, why even care about exponents? You will soon know why! Understanding any complex system such as income inequality or earth quakes require an understanding of power laws. Let's take a look at power laws in mathematical terms to get a better understanding:
Y is a function, X is the variable you can change, N is the exponent that you can scale and M is a constant that does not change.
We'll explore two types of relationships— linear and non-linear relationships.
To double a chocolate recipe, you would need twice as much as cocoa. This is simply a linear relationship.
Let's assume M (constant) is equal to 1 and N (exponent) is equal to 2. Ignore M here because anything you multiply with 1 will yield the original number. So let's plug some numbers in the equation (Y= X^N) and the Y will yield the following:
- If X is equal to 1, Y yields to 1
- If X is equal to 2, Y yields to 4
- If X is equal to 3, Y yields to 9
You get the point. A small change in the value of X leads to a proportionally large change in the value of Y. In the above example Y grows linearly. It requires twice the amount to make something twice the big.
How about non-linear relationships? It is much different because of complex systems. Let's use the most successful Olympian of all time, Michael Phelps, to understand non-linear relationships.
Le't take the same power law equation we explored earlier.
Michael Phelps has a low and a constant M because of some combination of his natural ability to swim and training history. X, the base is a number he has a control over, and N, the exponent is between 0 and 1. Thus the relationship between X and Y becomes less proportional. Let's plug some numbers in now.
- Y yields to 2 when N is 0.5, X is 4 and M is 1
- Y yields to 4 when N is 0.5, X is 16 and M is 1
Let's compare the two scenarios from above and see how it translates into his performance and training. In the first scenario above, Michael Phelps swims 4 extra miles (X variable for which he has control over) during his training which leads to an endurance of 2 (Y). In the second scenario, he has to swim 4 times more (16 miles) to double his endurance from 2 to 4.
As you increase X, you are likely to see an exponent of N decline. So increasing X of 16 to 64 is unlikely to double the endurance again. Eventually, the ratio of swimming extra lanes to an endurance will become nearly infinite. This is called diminishing returns— no matter how much more training time you put in, it will yield to a negligible or less positive result.
What happens when an exponent of N is negative? Le't take a scenario where Michael Phelps is injured.
- Y results in 0.5 when X is 4 and N is -(0.5)
- Y results in 0.25 when X is 16 and N is -(0.5).
In the first scenario, swimming 4 extra miles leads to only 0.5 a mile of endurance. In the second scenario that progress shrinks by half again. This wouldn't be smart way to train for Michael Phelps because more training leads to less endurance. This is an inverse relationship between the exponent X and Y.
Power law is complex to understand, but once you understand it becomes easy to see the world as is.
The greatest shortcoming of the human race is our inability to understand the exponential function. — Albert Allen Bartlett
Higher order power laws
We already know, when N is 1, it is called a linear relationship, also called a first-order power law.
When N is 2, it is called a second-order power law. Commonly found in physics concepts such as kinetic energy.
When N is 3, it is called a third-order power law. This law is applicable in wind turbines.
When N is 4, it is called a fourth-order power law. This law is applicable in heat radiation.
We don’t live in a normal world; we live under a power law. — Peter Thiel
As per everything else, power law should not be a singular lens to see the world. Like everything else, it requires a detailed analysis. As explained earlier, Michael Phelps cannot just rely on putting in more time to get a small percentage of endurance. Power laws have limitations such as law of diminishing return. The exponent starts declining in some applications, no matter how much extra time you put in.
There is a perfect example of diminishing return to happiness when you make more money. Most people assume making more money will bring more happiness. But that is not the case. Being able to enjoy basic needs will certainly bring happiness. So, earning from $20k to $60k will bring in joy, but earning from $100k to $115k leads to little change in well-being.
Power law is applicable in many systems, especially complex systems. Below are just to name a few:
- Compounding (Business): Another form of power law is called a compound interest. Small amount of contribution becomes a large amount of money. This is applicable to savings and debt.
- Zip's Law (Language): Another form of power law is called Zip's law which states a small percentage of words make up the majority of usage. The most used word in a language has a higher percent of all words used, while the second is used half as much, and so on. You can use Zip's law to understand several languages by only focusing on the most used words.
- Klieber's Law (Biology): The average lifespan of an animal based on its size can be attributed to power law. The larger the animal, the higher the lifespan, and vice-versa. The relationship between animal's size and its metabolism is identified as Klieber's law which states that an animal’s metabolic rate increases at 3/4ths of the power of the animal’s weight. Larger animals require less energy to burn their food compared to smaller ones which require more energy.