Introducing Power Laws - Amazon S3 · Summary of Unit Four Introducing Power Laws Introduction to Fractals and Scaling David P. Feldman

Summary of Unit Four

Introducing Power Laws

Introduction to Fractals and Scaling

David P. Feldman

http://www.complexityexplorer.org

David P. Feldman Introduction to Fractals and Scaling http://www.complexityexplorer.org

Initial Observations

● Box-Counting: ● If equation is true, there is self-similarity,

and we see a line on a log-log plot.● Reverse logic: If we see linear behavior

on log-log plot, there must be self-similarity.

● Power law:


Example: Word Frequencies in Moby Dick

● Determine frequencies of all words.● Plot histogram of frequencies.● There are 18,855 different words. There is one word that appears 14,086

times. There are 9161 words that appear only once.

● Data from: http://tuvalu.santafe.edu/~aaronc/powerlaws/data.htm


Normal Probability Disribution

● Average a, standard deviation sigma

● Strong central tendency.

● Small range of outcomes

● Fig source: https://explorable.com/normal-probability-distribution


Central Limit Theorem

● Let X_i be a set of random variables● Then the sum of N such random variables is

normally distributed as N gets large...● Regardless of the distribution of X_i.● (Provided that the distribution of the X_i's has

finite variance.)● A variable that is the result of a number of

additive influences should be normal.


Exponential Distribution● Also called geometric

distribution● Large range of

outcomes, but probability decreases very quickly

● Waiting times between events that happen with constant probability.


Power Laws have Long Tails●Power laws decay much more slowly than exponentials.

●Very large x values, while rare, are still observed.●Exponential: p(50) = 0.00000000078●Power Law: p(50) = 0.000244


Power Laws are Scale Free

● Power laws look like the same at all scales.


Exponentials are not Scale Free

● Exponential functions do not look the same at all scales


Power Laws are Scale Free

● Power Law

● Same ratio no matter what x is.

● Exponential

Ratio depends on x


Some Mathematical Notes

● Power laws are the only distribution that is scale free

● Discrete and Continuous probability distributions are different mathematical entities and need to be handled differently.

● However, from “20,000 feet” the difference is usually not crucial.


Power Laws and Averages

● Non-power law example: toss coin, get $1 if Heads, $0 otherwise.

● Average winnings quickly approaches $0.5.


St. Petersberg Game

● Keep tossing a coin until you get Heads. You win 2x, where x is the number of tosses.

● Rarely you get very large payoffs.● The average winnings does not exist: it is

infinite.


St. Petersberg Game



● the average does not exist.● the standard deviation does not exist.● Ex:



● the average does not exist.● the standard deviation does not exist.● Ex:


Power Laws

● Long tails.● Self-similar.● Sometimes averages or standard deviation

does not exist.● Very different from most distributions we're

used to.

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Introducing Power Laws - Amazon S3 · Summary of Unit Four Introducing Power Laws Introduction to Fractals and Scaling David P. Feldman