Some Common Distribution Curves for Growth and Decay:
Curves of Increasing Growth Curves of Decreasing Decay
Variable x is some underlying quantity that varies,
and constants A, B, r affect how y is dependent on x. Only the power-law
coefficient alpha (α)
is scale free in the sense that it is not affected by multiplication or division of x by an
arbitrary constant. This means that no matter how you measure interval variable x,
if there is a power-law relationship between x and y, you will always get the same alpha.
Power-laws thus define universality classes defined by alpha. For example
see: West, G.B., Brown, J.H. & B.J.Enquist (1997) A general model for the origin of
allometric scaling laws in biology. Science 276:122-126. (PDF Reprint)
Linear distributions (green lines) are straight lines in ordinary unlogged scatterplots (Figure 1).
In general y=A + Bx where A is an initial value and a positive B is an additive growth constant while a negative B is a subtractive decay constant.
Exponential distributions (pink dotted lines) are linear only in semi-log plots (Figure 2).
In general, applied to an initial value A, a fixed rate r of growth or decay is raised to successively higher powers (x), with result y = A rx, hence log y = log A + r x (the log of y varies with x). In the continuous case, y=Aek x (hence ln y = ln A + k x), where e=2.718 is the base constant of the natural log, denoted ln, and k=ln(r) is the natural log of r, the power to which e must be raised to get r. In the examples y = 1 e.805 x and y = 25 e-.805 x = 25/e.805 x. In the exponential the base (A or e) is constant while the exponent changes with x. Hence the curve starts with slow growth and accelerates with x.
Power-Law or Pareto distributions (purple lines) are linear only in log-log plots (Figure 3).
In general y=Axα (hence ln y = A + α ln x) where A is an initial value and α is the power constant. In the examples y = 1x2 and y = 25x-2 = 25/x2. In the power law the base (x) changes while the exponent (α) is constant. Power-law exponents usually go no higher than 3 (for growth) or lower than –3 (for decay). The power law rises or falls more quickly at lower values of x compared to the exponential but changes more slowly at higher values, hence 'fat tail' decay.
The symmetries in the graphs are created by operations that reverse one another, called duals:
a. addition and subtraction by a constant are duals in linear distributions.
b. multiplication and division by a constant are duals in exponentials;
c. positive and negative exponentiation by a constant are duals in power-laws (positive exponents may be added, negative subtracted).
A number of asymmetries derive from the fact that power-law dynamics are relative to where you are at on the x axis raised to a fixed power, while exponential dynamics are relative to a fixed starting value raised to a power that varies with the x axis. Thus, for example, exponential population growth occurs faster than power-law growth as a function of time: its hi-rise occurs later. In the plots, exponentials are the lower curve in growth but the middle in decay; power laws are the lower curve in decay but the middle in growth. The gray areas in the ordinary (top) scatterplots show the 'fattest' distributions: the slow-start and hi-rise distributions for exponential growth and the fast-drop and fat-tail distributions for power-law decay.