This morning, before work, I finished reading *Connected: The
Surprising Power of Our Social Networks and How They Shape Our Lives*.
The book was written by two social scientists with PhDs. And in one
paragraph they wrote the following (emphasis mine).

There is only one relationship between two people, but there are three possible relationships between three people, six between four people, ten between five people, and so on. Since the number of possible relationships growsexponentiallywith group size, it probably takes a big shift in cognitive capacity to keep up with all the drama of a full social life.

I'm not sure what it takes to be a professor at Harvard or one at UC San
Diego, but *my* readers will understand the difference between
*exponential growth* and *polynomial growth*. And I'm going to
explain it in an way without resorting to formulae because this is
actually quite simple.

Exponential growth means that the rate of growth is proportional to the size of the thing growing. Simply put, the growth rate may be described as an unchanging percentage rate. For example, a fixed-rate savings account grows exponentially because it accrues interest at the same percentage rate (e.g., 1.5% per year) regardless of how much money is in the account.

Now, take the sequence of growing relationship numbers in the above
*Connected* quote: 1, 3, 6, 10, (15, 21,) … What is the
percentage increase between any two consecutive elements? It depends on
which elements. From 1 to 3 is a 200% increase. From 3 to 6 is a 100%
increase. From 6 to 10 is a 67% increase. From 10 to 15 is a 50%
increase. And so on. It is not a flat rate, which means that the rate of
growth is not proportional to the size of the thing growing. Thus, it is
not exponential. But what is it?

If something is growing fast but is not growing exponentially, then it
is probably growing polynomially. This is a good rule of thumb if
you're dealing with “real world” sorts of numbers. Two of the most
common types of polynomial growth are *linear growth* and
*quadratic growth*. Linear growth is any sequence in which the
increase between any two consecutive elements is the same size as the
increase between any other two consecutive elements. For example, the
sequence 6, 8, 10, 12, 14, … is linear.

Quadratic growth means that the increase between any two consecutive
numbers is the same as the previous increase, plus or minus some other
constant number. So, take again our *Connected* numbers: 1, 3, 6,
10, 15, 21, … What is the *size* of increase between any two
consecutive elements? It depends on which elements. From 1 to 3 is an
increase of 2. From 3 to 6 is an increase of 3. From 6 to 10 is an
increase of 4. From 10 to 15 is an increase of 5. And so on. In this
case, each increase is exactly +1 larger than the previous increase.
This is the hallmark of quadratic growth and will form (half of) a
parabola if you graph it, but I'll save that for traditional math
classes.

There are other forms of polynomial growth, but in the real world, the two types linear and quadratic will cover most cases that aren't exponential. And now you know.

*If the rate of growth cannot be described as a flat percentage
rate, then it is* not *exponential growth*.

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