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
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