Some Common Distribution Curves for Growth and Decay:

The graphs for examples were done with Excel Spreadsheet (here the exponential r = √5)

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 r^{x}, hence log y = log A + r x (the log of y varies with x). In the
continuous case, y=Ae^{k 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=Ax^{α} (hence ln y = A + α ln x)
where A is an initial value and α is the power constant. In the
examples y = 1x^{2} and y = 25x^{-2}
= 25/x^{2}. *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.