Growth

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 grows exponentially with 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.