Power-laws
When I first began studying complex systems, I did not understand why scientists are so interested in power-laws. Particularly in physics, where there are while papers that show a phenomenon follows a power law and they give a rough estimate of the slope. The goal of this page is to give a simple introduction to why power-laws are so interesting.
For those who are wondering what a power-law is, I will give a quick introduction
Mathematically, a power-law is any relation having a form
$$y=a x^b \hspace{0.5in} \mathrm{OR} \hspace{0.5in} y \sim ax^b$$where ~ means that for large values of x, y approaches the given form.
A familiar math example of power laws were the exponent b is an integer are the polynomials.
$$ y = \sum_{n=0}^{N}a_{n} x^n \sim a_{N}x^{N}$$Power-laws are typically more general in that the exponent b can be any real number.
Some simple geometrical examples of power laws are areas, circumference, and volumes of regular shapes.
$$ \mathrm{Circle} $$ |
$$ \mathrm{Square} $$ |
|
$$A = \pi r^2$$ |
$$ V = \frac{4}{3}\pi r^3 $$ |
|
$$C = 2 \pi r$$ |
$$A = 4 \pi r^2$$ |
The easiest way to tell is some phenomenon follows a power law distribution is to plot the corresponding relationship on a log-log plot. If the relationship forms a straight line, the relationship is a power law with exponent equal to the slope.
The reason why we get a straight line can be seen by taking the logarithm of both sides of the algebraic form of a power-law.
$$\log\left(y\right) = \log\left(a x^b\right) = \log\left(x^b\right)+log\left(a\right)$$ $$ = b \log\left(x\right) + \log\left(a\right)$$Comparing to y=mx+b we see that the exponent gives the log-log plot slope
Often we we say that a phenomenon follows a power-law, we typically mean that some aspect of the phenomenon follows a power-law distribution.
Power-law distributions have a simple interpretation of being related to the ration of the number of events of a certain size to the number of events a size smaller. For example, earthquakes follow a power-law. distribution with a slope of 1, there are 10 times as many earthquakes 1/10 the size or alternatively there are 1/10 the number of earthquakes 10 times as large.
There are many natural phenomenon that follow power-law distributions: earthquakes, avalanches, turbulence, epidemics, market fluctuations, etc.
So why are we interested in power-laws. There are many reasons, but here is a short list.
Power-laws typically indicate criticality and universality.
Criticality and is a property associated with critical points. It is typically found in second order phase transitions. One of the main aspects of criticality is that many properties of the system diverge as a power-law.
Universality encapsulates the observation that the statistical properties of large classes of systems are identical, independent of the dynamic details of the system. Initially, there was some hope that natural phenomenon could be characterized by a relatively small number of universality classes.
There are well developed methods for understanding criticality and universality which can be used if a phenomenon has power-law behavior.
No Characteristic Size and Scale-invariance
One of the really amazing properties of power-laws is that the lack a characteristic size. To see what I mean by this, consider one of the geometrical examples introduced earlier: a sphere. How big is a typical sphere? This question does not really have an answer because there is no such thing as a typical sphere. The Earth looks exactly like an epic sized ball bearing. Similarly there is no such thing as a typical sized epidemic. The progression of infectious disease within a household is nearly identical to the progression of an infectious disease within a city.
This epidemic gives a good window into the property of scale-invariance. The basic idea is that a phenomenon looks the same regardless of what scale you look at it. A nation is made of cites, cites are made of neighborhoods, neighborhoods are made of households, and the progression of infectious disease at all scales is nearly identical.
Scale-invariance is typically what is meant when a phenomenon is called fractal.
Use Small to Study Large
The lack of a characteristic size and scale invariance means that the same physics applies regardless of the scale of the system. This means that by studying the physics of small systems you are studying the same physics that applies in large systems. For example, earthquakes have been observed to have scale-invariance and so results from rock fracture experiments in the lab can be applied to earthquakes.
One area where scale-invariance is particularly useful is when there are not enough large events to obtain good statistics. If the phenomenon follows a power-law distribution, the statistics of small events are identical to the statistics of large events. Because there are many more small events, statistics can be obtained using the small events and then those statistics can be used for large events.
Risk of Very Large Events
Power-law distributions produce more large events than pretty much any other distribution.
This means that the time between large events is much shorter than events with exponential time dependence (diffusion, radioactive decay, etc.). One important thing to remember is that the typical arguments of a process taking longer than the lifetime of the universe do not apply to power-laws.
Additionally, in systems that follow power-laws, the dynamics do not place a limit on the largest possible event. The only constraint is the size of the system, so the largest possible event is an event on the scale of the system. This is very important considering epidemics and stock market crashes follow power-laws. This means the largest epidemic is a global epidemic and the largest stock market crash is one in which nearly all value is lost.
The short time between large events and the absence of a largest event are very important when trying to assess the risk associated with events that follow power-law distributions.
This is just a short list of why power-laws receive so much attention.