Power Law Distribution

A Power Law Distribution, also sometimes called a Pareto Distribution, is a statistical quirk that describes how certain types of data are spread out. It’s different from the familiar bell curve you might see in normal distributions. The power law can be used to describe a phenomenon where a small number of items is clustered at the top of a distribution (or at the bottom), taking up 95% of the resources. In other words, it implies a small amount of occurrences is common, while larger occurrences are rare.

Key Points:

  • Tail Heavy: Power law distributions have a “fat tail” on the right side. This means there’s a higher probability of encountering extreme values (large pebbles) compared to a normal distribution.
  • Scale Invariance: Often, power laws exhibit scale invariance. This means that if you zoom in or out on the data, the overall shape of the distribution remains similar. Imagine looking at pebbles on the beach from afar versus close up – the distribution of sizes might look similar.

Y = c * X ^ α

  • Y is the frequency or probability of something happening (like the number of pebbles of a certain size).
  • X is the size of something (like the diameter of a pebble).
  • α (alpha) is the exponent, which tells you how much the frequency changes as the size changes.
  • c is a constant that scales the whole thing.

Power law distributions describe situations where a small number of things are very common, and larger things, while less frequent, still hold significant weight. They are a handy tool for understanding how some types of data are skewed towards extremes.