Ever heard of a magic bean that doubles every second? Yes, in your Algebra class! Consider you got one such bean and you want to fill a huge bowl. How much time will it take to fill 5% of the bowl? Maybe hours! But once it reaches 5% of the bowl, how much time will it take to completely will the bowl? Just 5 Seconds! Yes. Since it doubles every second, the 5% would double to 10%, 10 to 20, 20 to 40, 40 to 80.. and by the end of the 5th second, you would already be overflowing your bowl. The sorcery of these magic beans is explained by what we call as the Power Laws! .
What are power laws?
Power laws describe non-linear relationships between two quantities. They are called power laws because one quantity can be explained as the power of another quantity. This could be represented as
y = a∗ 𝒙𝒌
where a is a coefficient and k is the exponent of the power law. They capture relationships wherein a small change in one variable can lead to a big change in the other variable.
A simple example of a power law
One example of a power law is analyzing a cube by looking at the relationship between its edge length and volume. The volume of a cube is the cube of its edge length, expressed by the formula:
volume = edge_length3
The volume of a cube of edge length 1 would be 1 but the volume of a cube with edge length 2 would be 8. Thus, a 2x increase in edge length leads to an 8x increase in the volume. Similarly, a 100x increase in the edge length would lead to a 1,000,000x increase in the volume.
Different order of power laws
Different orders of power laws exist based on the value of the exponent. Below are some examples:
1st order. These are the power laws where the exponent has a value of 1. These laws cover linear relationships between variables. Consider a case where you are renting a taxi and the taxi charges $4 + $1.50 for every kilometer. This cost equation can be captured using the following equation:
cost of taxi ride = 1.5 ∗ distance covered (in Km) + 4
2nd order. These are the power laws where the exponent has a value of 2. The celebrated theory of relativity is captured using a 2nd order power law where energy equals mass times the speed of light squared.
e = m ∗ c2
3rd order. These are the power laws where the exponent has a value of 3. The equation of the volume of cube is an example of the third order power law.
volume = edge_length3
Similarly, there exist higher power laws that capture many complex non-linear relationships.
Some powerful manifestations of power laws around us
Power laws manifest in various forms around us. From the building blocks of our universe such as gravity and electromagnetism where the magnitude of force depends on the inverse of the square of the distance, to the modern computer networks where Metcalfe’s law dictates the value of a telecommunications network, and everything in between such as empirical laws indicating wealth inequalities and population distributions. Below are some celebrated examples:
Moore’s law: One such famous power law is Moore’s law, which was postulated by Gordon Moore, co-founder of Fairchild Semiconductor and Intel. The law states that the number of transistors on a microchip would double every two years while its operating cost would drop by half. The law gives us the equation to assess future processing power. If Ct is the number of transistors in the current year, Ft is the number of transistors in the future, and k is the difference between the future and current year, then:
Ft = Ct ∗ 2k/2
Law of Gravity: Another power law is the law of gravity. All objects with a mass attract each other. We can find the impact of this force (F) with the following formula:
Here, m1 and m2 are the masses of both the objects respectively, r is the distance between them, and G is the universal gravitational constant.
Compound interest: The formula for compound interest is a power law that forms the backbone of our entire financial system. This concept of earning interest on previous interest makes the total amount increase non-linearly with time. I think this is where the phrase “it takes money to make money” was born! If P is the principal amount, i is the rate of interest, and t is the number of interest periods, then the interest earned can be calculated using the following power law formula:
Interest earned = P ∗ ((1+i)t − 1)
Power laws in ignio
Power laws manifest in various forms in the underlying analytics framework of ignio as well. ignio uses multi-variate regression models to capture relationships between various metrics in the form of power-law equations. These models are then used to profile normal behavior, detect anomalies, predict future behavior, and assess impact of change.
Power laws are all around us, manifesting themselves in a variety of ways. They are the unseen forces that govern nearly every aspect of our lives. From the relationship between a planet’s orbit and its revolution time to the population distribution of the cities in which we live; from how strong gravitational forces would attract us to how much food we will require for survival; and from calculating the optimal speed for driving on a curved road based on its degree of inclination to calculating the required speed of satellites to keep them in orbits. Just imagine our lives if these laws were never discovered. Most likely, there are many more such laws still waiting to be discovered to revolutionize the way we see our world.