## Wednesday, February 28, 2018

### Power laws

Definitions

A power law can be defined as ”A relationship between two quantities such that one is proportional to a fixed power of the other”. It can also be defined as ”A relative change in one quantity results in a proportional relative change in another”.

Basics about the scales of power law

When you find out what the value of the power is in the power law, you can see how things scale. When the power is 1, dependent variable doubles, when the independent variable doubles. When the power is more than 1, the dependent variable more than doubles. For example, when the power is 2, dependent variable quadrubles, when the independent variable doubles. If the power is below 1, then the dependent variable is less than a doubling independent variable.

Power laws in nature

You can see many power laws working in nature. For example, animals in the world are roughly cube shaped. The amount of skin goes up by the factor of 4, when the surface area (m²) is doubled. In the same time the volume (m³) of guts goes up by a factor of 8. Animals lose heat through their skin and generate it through their guts. When the animals grow, their bone structure has to change as the bones´ ability to support the animal grows slower than the weight of the animal. Compare square with a cube. What this basically means that animals with lots of weight need to have strong bones to support it. For example, an elephant has thick legs, because they carry a lot of weight.

The metabolic rates of animals scale to their mass to the ¾ power. It tells you that the amount of the energy you need to survive. It basically means that the larger you are, the less energy you need compared to your per unit of mass. These kind of scaling laws apply to the oxygen intake and heart rate. Breathing and heart rates scale with mass to the – ¼ power. These power laws are approximations. They do not give exact truths. They are estimates based on existing data. These power laws tell about physical limits of each living thing. You cannot find a giant with thin legs or live forever.

Power laws in human constructions like cities

These laws apply to human behavior like wars and human constructions like languages and cities. For example, Lewis Fry Richardson found out a power law about wars. The higher the number of people killed in the war, the more time it takes to a war as destructive as the former one. You can also find same kind of power law working in languages. How often you use a word is described by an exponent of -1. The word that is the second most common in language is found half the times compared to the most common word. This power law applies also the population of cities in different countries and the sizes of companies. The largest city in the country has approximately twice the population of the second largest city. There are many power laws found in city sizes. For example, the bigger the population of the city, the less each person uses gasoline and the less road they need. All these power laws can be used in planning future cities.

Pareto´s law

Pareto´s law means that the minority of causes, inputs or efforts usually lead to a majority of the results, outputs or rewards. This rule states that there is an in-built imbalance between causes and results, inputs and outputs. Sometimes, this rule is defined as a 20/80-rule. For example, 20% of the customers bring 80% of the profits for the company. I wouldn´t really describe Pareto´s law with 20/80 rule, because these scales have so much variation. For example, only a small percentage of the authors sell almost 100% of the books. You have to invert, when you are using Pareto´s law too. For example, you need to understand that most of the bad things that happen have only few reasons. Most of your losses come from few sources. These rules apply to your relationships too. Most people are insignificant for your well-being, but few people cause most of the bad things in your life. Pareto´s law also means that small changes or lucky events may give big results. For example, when Oprah has recommended some products, their sales have gone through the roof.

Universe is full of power laws. You should understand these non-linearities as well as linear cause-and-effect relationships. These laws are much more usual than everybody thinks. Most of the people never notice them. Using power laws to your advantage is hard, but the results are worth the effort. Most of the misunderstood risks come from power laws. Do not ignore them.

Sources:

These power laws are related to misunderstood risks. They will be the topic for the next week.

-TT