Power law

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Power laws[1] describe relations where one value changes as if raised to a constant exponent.

For example, length and area are related in a L² way, length and volume are related in a L³ way.

The term 'power law' tend to be used when modeling or describing

When trying to model such things, power laws help describe the somewhat-more-statistical properties, e.g. the fairly specific falloff of probability.

There are many everyday things (physical, biological, man-made or natural), that fall off in such ways, from

frequencies of family names the species richness in clades of organism

The reasons behind these vary - some are relate to growth, some to self-reinforcement, structural redistribution, some indirectly to area and volume effects.

Examples:

the distance taken in foraging - usually small, but occasionally larger out of necessity, and while relatively rare they are part of the pattern, not outliers.

sizes of the craters on the moon - most impacts are small, a few large (mostly following asteroid size, itself following power law)

neuron activity patterns

the frequencies of words in most languages

the frequencies of family names

the species richness in clades of organisms

human judgments of stimulus intensity

The reasons behind it may be part of a complex system, that we can only describe the surface of.

the sizes of power outages - small ones are common, large ones are rare

sizes of clouds - small ones are numerous, large ones are rare

the sizes of volcanic eruptions - small ones are common, large ones are rare

This is sometimes interesting in itself - say, why are volcanic eruption sizes not gaussian but lower-law?

Note that there are many cases where a pure power law doesn't quite fit empirical data, because it suggests values for arbitrarily large and small values.

Which may not make real-world sense. Say, a a forest fire has only so many trees to consume.

So in modelling or other numerical methods, it can make more sense to use a truncated power law, which puts a maximum (and/or minimum) on the values that can be taken.

(In statistics) See e.g. include the Zipf distribution[2], zeta distribution[3], Pareto distribution[4], and a few others.

(e.g. 'scale free distribution' in scale-free networks describes a similar idea)

Note that these often do not satisfy more stringent statistical properties -- see also Statistics_notes_-_on_random_variables,_distributions,_probability#Less_formal_descriptions.

Other things that come to mind: Stevens's power law, the idea that most stimuli need a magnitude increase to be felt roughly linearly more strongly.

Zipf's law, Zipfian word distributions

Linguists often specifically know Zipf's law, which refers to to the observation that when you count words, not only are a few very common and a lot fairly rare, for most of the words it roughly holds that each word's frequency in a text is roughly inversely proportional to its rank.

There are various ways to graph this, see e.g. the graphs in the various references.

This has implications such as that in top so-many terms in some text accounts for the bulk of that text. In corpora of text, half the word use is often covered by the top 200 or so words that occur in that text.

In languages that have function words (which is many of them), those are likely to take most or all places in the top ten while carrying near-zero semantic value (more, but still relatively little, once you consider syntax)

For example, you may find that in some English documents

the (rank 1) occurs as ~7% of all words
of (rank 2) occurs ~3.5%
and (rank 3) occurs ~2.9%
You can estimate that something at rank 10 would occur ~0.7%
something at rank 1000 occurs 0.007%
...etc.

This largely holds to when you e.g. do n-grams or otherwise use larger units.

For example, if you analyse emails or some chat logs, or just people interacting, the top sentences by count consist largely of formalities, interjections, and (other) daily social interactions.

Power law

Zipf's law, Zipfian word distributions

See also

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