🌌Power Laws

Almost everything follows power laws.

What is a Power Law?

A power law is a mathematical relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other, independent of the initial size of those quantities. The general form of a power law is:

$$y = Ax^n$$

Here, $y$ and $x$ are the variables, $A$ is a constant, and $n$ is the exponent.

This type of relationship suggests that the change in one variable leads to a predictable change in the other, following a consistent pattern.

Normal Distributrion Vs Power Law Distribution

Simple Explanation

In simpler terms, a power law describes situations where a specific change in one factor causes a corresponding change in another.

For example, if doubling the number of users on a social media platform results in a fourfold increase in network value, this relationship can be described by a power law.

Similarly, if increasing the length of a side of a square leads to an increase in the area by the square of the length change, it follows a power law.

Log-Log Plots

Log-log plots can be used to visualize power laws.

Log-log plots can help make these mathematical relationships easier to understand and identify.

Consider the power law $y = Ax^n$ again.

Taking the logarithm of both sides, the power law becomes:

$$log (y) = alog (x) + log (b)$$

This is a linear equation (in the logarithmic space) where $a$ is the slope and $log (b)$ is the y-intercept.

Therefore, a power-law relationship will appear as a straight line on a log-log plot (such as Figures below), with the slope of the line corresponding to the exponent in the power law.

y_Linear - x_Linear Plot Visualization

y_Log - x_Log Plot Visualization

Examples of Power Laws in Nature and Other Contexts

Planetary Orbits

The Third Law of Kepler states that the square of the orbital period $T$ of a planet is proportional to the cube of its average distance $a$ from the Sun:

$$T^2 \propto a^3$$

Log-log plot of the orbital period versus average distance from the Sun for planets in the Solar System, illustrating Kepler's Third Law.

The regression line fits the planetary data to high accuracy, as shown by the scatter plot on a log-log scale.

Gravity

The force of gravity between two objects follows a power law.

Newton's Law of Universal Gravitation states that the gravitational force $F$ between two masses is proportional to the product of their masses ${{m_1 m_2}}$ and inversely proportional to the square of the distance $r$ between them $r^2$ following a power law with an exponent of -2.

$$F = G \frac{{m_1 m_2}}{{r^2}}$$

This is a classic example of a power law, where the force decreases with the square of the distance.

Earthquakes

The frequency and magnitude of earthquakes follow a power law distribution.

The Gutenberg-Richter law describes that the number of earthquakes decreases exponentially with increasing magnitude.

Earthquake Statistics Gutenberg-Richter relation

This means that small earthquakes are much more common than large ones.

Population and City Sizes

In social sciences, the distribution of city sizes often follows a power law.

This means that there are a few very large cities and many smaller towns.

Following Auerbach's 1913 observation, there has been substantial examination of Zipf's law for city sizes.

The same pattern is observed in the distribution of population sizes across countries.

Wealth Distribution

In economics, power laws describe the distribution of wealth.

A small percentage of the population controls a large portion of the wealth, while the majority controls a smaller fraction.

This is often visualized with the Pareto Principle or the "80/20 rule."

Biological Systems

In biology, power laws can describe various phenomena, such as the distribution of sizes in animal populations, the branching patterns of trees and blood vessels.

Metabolic Rate - Log-Log visualization

For example, the relationship between an animal's body weight and its metabolic rate in kilocalories, where larger animals tend to have a slower metabolism per unit of body mass compared to smaller animals.

Why Are Power Laws Important?

Power laws are important because they help us understand and predict the behavior of complex systems.

They reveal underlying patterns that are not immediately obvious and are found across a wide range of disciplines, from physics to economics.

By understanding these patterns, we can better comprehend the structure and dynamics of the world around us.

Conclusion

Power laws are a fundamental concept that appears in various natural and social phenomena.

Examples of the power law distribution. Adapted from: M. Newman, Power laws, Pareto distributions and Zipf’s law, Contemporary Physics, 2007.

They provide a framework for understanding how different quantities are related and how changes in one can affect another.

Whether in the forces governing planetary orbits or the distribution of wealth, power laws offer valuable insights into the world we live in.