Post on 15-May-2020
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:
David P. Feldman Introduction to Fractals and Scaling http://www.complexityexplorer.org
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
David P. Feldman Introduction to Fractals and Scaling http://www.complexityexplorer.org
Normal Probability Disribution
● Average a, standard deviation sigma
● Strong central tendency.
● Small range of outcomes
● Fig source: https://explorable.com/normal-probability-distribution
David P. Feldman Introduction to Fractals and Scaling http://www.complexityexplorer.org
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.
David P. Feldman Introduction to Fractals and Scaling http://www.complexityexplorer.org
Exponential Distribution● Also called geometric
distribution● Large range of
outcomes, but probability decreases very quickly
● Waiting times between events that happen with constant probability.
David P. Feldman Introduction to Fractals and Scaling http://www.complexityexplorer.org
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
David P. Feldman Introduction to Fractals and Scaling http://www.complexityexplorer.org
Power Laws are Scale Free
● Power laws look like the same at all scales.
David P. Feldman Introduction to Fractals and Scaling http://www.complexityexplorer.org
Exponentials are not Scale Free
● Exponential functions do not look the same at all scales
David P. Feldman Introduction to Fractals and Scaling http://www.complexityexplorer.org
Power Laws are Scale Free
● Power Law
● Same ratio no matter what x is.
● Exponential
Ratio depends on x
David P. Feldman Introduction to Fractals and Scaling http://www.complexityexplorer.org
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.
David P. Feldman Introduction to Fractals and Scaling http://www.complexityexplorer.org
Power Laws and Averages
● Non-power law example: toss coin, get $1 if Heads, $0 otherwise.
● Average winnings quickly approaches $0.5.
David P. Feldman Introduction to Fractals and Scaling http://www.complexityexplorer.org
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.
David P. Feldman Introduction to Fractals and Scaling http://www.complexityexplorer.org
St. Petersberg Game
David P. Feldman Introduction to Fractals and Scaling http://www.complexityexplorer.org
Power Laws and Averages
● the average does not exist.● the standard deviation does not exist.● Ex:
David P. Feldman Introduction to Fractals and Scaling http://www.complexityexplorer.org
Power Laws and Averages
● the average does not exist.● the standard deviation does not exist.● Ex:
David P. Feldman Introduction to Fractals and Scaling http://www.complexityexplorer.org
Power Laws
● Long tails.● Self-similar.● Sometimes averages or standard deviation
does not exist.● Very different from most distributions we're
used to.