Fractals, Power Laws, and Scale-Free Distributions
A power-law tail follows P>x = K x^-α: multiplying the threshold by a given factor changes its exceedance probability by a fixed proportion. This scaling property links power laws to fractals, but identifying a power law in data and deciding where it applies remain separate problems.
THE IDEA
FRACTALS, POWER LAWS, and SCALE-FREE DISTRIBUTIONS
Definition: Technically, P>x= K x -α where α is supposed to be the power-law exponent. It is said to be scale free, in the sense that it does not have a characteristic scale: relative deviation of 
does not depend on x, but on n—for x “large enough.” Now, in the other class of distribution, the one that I can intuitively describe as nonscalable, with the typical shape p(x) = Exp[-a x], a is the rate parameter and its inverse sets the characteristic scale.
Problem of “how large”: Now the problem that is usually misunderstood. This scalability might stop somewhere, but I do not know where, so I might consider it infinite. The statements very large and I don’t know how large and infinitely large are epistemologically substitutable. There might be a point at which the distributions flip. This will show once we look at them more graphically.
Log P>x = -α Log X + C for a power-law tail. When we do a log-log plot (i.e., plot P>x and x on a logarithmic scale), as in Figures 15 and 16, we should see a straight line.
Fractals and power laws: Mandelbrot (1975, 1982). Schroeder (1991) is imperative. John Chipman’s unpublished manuscript The Paretian Heritage (Chipman [2006]) is the best review piece I’ve seen. See also Mitzenmacher (2003).
“To come very near true theory and to grasp its precise application are two very different things as the history of science teaches us. Everything of importance has been said before by somebody who did not discover it.” Whitehead (1925).
Fractals in poetry: For the quote on Dickinson, see Fulton (1998).
Lacunarity: Brockman (2005). In the arts, Mandelbrot (1982).
Fractals in medicine: “New Tool to Diagnose and Treat Breast Cancer,” Newswise, July 18, 2006.
General reference books in statistical physics: The most complete (in relation to fat tails) is Sornette (2004). See also Voit (2001) or the far deeper Bouchaud and Potters (2002) for financial prices and econophysics. For “complexity” theory, technical books: Bocarra (2004), Strogatz (1994), the popular Ruelle (1991), and also Prigogine (1996).
Fitting processes: For the philosophy of the problem, Taleb and Pilpel (2004). See also Pisarenko and Sornette (2004), Sornette et al. (2004), and Sornette and Ide (2001).
Poisson jump: Sometimes people propose a Gaussian distribution with a small probability of a “Poisson” jump. This may be fine, but how do you know how large the jump is going to be? Past data might not tell you how large the jump is.
Small sample effect: Weron (2001). Officer (1972) is quite ignorant of the point.
Recursivity of statistics: Taleb and Pilpel (2004), Blyth et al. (2005).
Biology: Modern molecular biology pioneers Salvador Luria and Max Delbrück observed large fluctuations in mutant counts across bacterial cultures, including occasional cultures with far more mutants than others.
Thermodynamics: Entropy maximization without the constraints of a second moment The two exhaustive domains of attraction: vertical or straight line with slopes either negative infinity or constant negative α. Note that since probabilities need to add up to 1 (even in France) there cannot be other alternatives to the two basins, which is why I narrow it down to these two exclusively. This compressed note does not state the assumptions needed for its two-domain claim and should not be read as an exhaustive classification of all distributions.
When to use it
Modeling the distribution of city sizes, where a few megacities far exceed the median.
Risk management in finance: comparing heavy-tailed models, including power laws, and examining how tail assumptions affect loss estimates.
Earthquake risk: using the Gutenberg–Richter relationship to describe the relative frequency of earthquakes of different magnitudes.
Genetics: modeling large fluctuations in mutant counts across bacterial cultures.
FROM THEORY TO EVERYDAY LIFE
In practice
Luria-Delbrück experiment on bacterial mutations
Salvador Luria and Max Delbrück observed wide variation in mutant counts across bacterial cultures, with occasional exceptionally large counts. The example concerns the number of mutants, not unusually large individual bacteria.
Wealth distribution and Pareto principle
A Pareto model can describe the upper tail of a wealth distribution, where a small share of people hold a large share of wealth. Its scaling property applies over the range being modeled; it does not mean that the entire population has no median or other measure of typical wealth.
A MOMENT TO REFLECT
Put it to work.
How can we empirically distinguish a true power-law distribution from a finite-sample exponential distribution?
In what real-world situations might the 'large enough' threshold for a power law be unknowable, and what are the practical consequences?
Why is it problematic to assume that past data can predict the size of a 'Poisson jump' in a Gaussian model?
Often confused with
References
- Fractals: Form, Chance and Dimension — Benoit Mandelbrot (1975)
- The Fractal Geometry of Nature — Benoit Mandelbrot (1982)
- Fractals, Chaos, Power Laws — Manfred Schroeder (1991)
- The Paretian Heritage — John Chipman (2006)
- A Brief History of Generative Models for Power Law and Lognormal Distributions — Michael Mitzenmacher (2003)
- Science and the Modern World — Alfred North Whitehead (1925)
- Dickinson's Fractal Imagery — Alice Fulton (1998)
- Lacunarity — John Brockman (2005)
- New Tool to Diagnose and Treat Breast Cancer — Newswise (2006)
- Critical Phenomena in Natural Sciences — Didier Sornette (2004)
- The Statistical Mechanics of Financial Markets — Johannes Voit (2001)
- Theory of Financial Risk and Derivative Pricing — Jean-Philippe Bouchaud and Marc Potters (2002)
- The Physics of Complex Systems — N. Boccara (2004)
- Nonlinear Dynamics and Chaos — Steven Strogatz (1994)
- Chance and Chaos — David Ruelle (1991)
- The End of Certainty — Ilya Prigogine (1996)
- On the Philosophy of the Problem of Power Laws — Nassim Taleb and Avital Pilpel (2004)
- On the Distribution of Earthquakes — Vladimir Pisarenko and Didier Sornette (2004)
- Calibration of the Power Law — Didier Sornette et al. (2004)
- The Power Law of Extreme Events — Didier Sornette and K. Ide (2001)
- On the Distribution of Extremes — Rafal Weron (2001)
- The Distribution of Stock Returns — R. R. Officer (1972)
- Recursivity in Statistics — Nassim Taleb and Avital Pilpel (2004)
- Power Law Distributions in Biology — Simon Blyth et al. (2005)
CONTINUE THE CONVERSATION
Source texts
- The Black Swan The Impact of the Highly Improbable With a new section On Robustness and Fragility (Nassim Nicholas Taleb)