Exponents & Power Laws
The term power law describes a functional relationship between two quantities, where one quantity varies as a power – or exponent – of another.1 Part of the definition to linear systems is that the relationship between input and output is related in a linear fashion. The ratio of input to output might be 2 times as much, 10 times as much or even a thousand times. It is not important, as this is merely a quantitative difference. What is important is that this ratio between input and output is itself not increasing or decreasing. But with feedback loops the previous state to the system feeds back to affect the current state, thus enabling the current ratio between input and output to be greater or less than its ratio previously, and this is qualitatively different.
This phenomenon is captured within mathematical notation as an exponential. The exponential symbol describes how we take a variable and we multiply it by another, not just once but we in fact iterate on this process, meaning we take that output and feed it back in to compute the next output. Thus, the amount we are increasing by each time itself increases. An example of this might be the growth of bacteria when given the right conditions. If I wish to create a test tube of bacteria, knowing that the bacteria will double every second, I start out in the morning with only two bacteria hoping to have my tube full by noon. As we know the bacteria will grow exponentially as the population at any time will feed into affect the population at the next second, like a snowball rolling down a hill. It will take a number of hours before our tube is just 3% full but within the next five seconds as it approaches noon it will increase to 100% percent of the tube. This type of disproportional change with respect to time is very much counter to our intuition where we often create a vision of the future as a linear progression of the present and past. The important thing to note here is that in exponential growth the rate of growth itself is growing, and this only happens in nonlinear systems where they can both grow and decay at an exponential rate.
Exponentials are also called powers and the term power law describes a functional relationship between two quantities, where one quantity varies as a power of another. There are lots of examples of the power law in action but maybe the simplest is the scaling relationship of an object like a cube. A cube with sides of length A will have a volume of A3, and thus the actual number that we are multiplying the volume by grows each time. This would not be the case if there was a simple linear scaling such as the volume being 2 times the side length. Another example from biology is the nonlinear scaling within the metabolic rate vs. size of mammals. The metabolic rate is basically how much energy one needs per day to stay alive, and it scales relative to the animal’s mass in a sub-linear fashion. If you double the size of the organism then you actually only need 75 % more energy. One last example of the power law will help to illustrate how it is the relationships between components within a system that is a key source of this nonlinearity.
The so-called Metcalfe’s Law comes from the world of I.T. and it derives from the simple observation that the number of possible cross connection in a network grows as a square of the number of computers in the network. Every time we add a new computer to the network we have the possibility of adding as many more links as there are computers in the network. So whereas the number of computers grows in a linear fashion the number of links can grow in a super linear or exponential fashion. As each person who joins the network makes it more valuable, Metcalfe’s law leads to the value or power of a network increasing in proportion to the square of the number of nodes on the network. This is of course not restricted to just computer networks, but is a feature of all networks and thus is given the more general name the network effect. The network effect is a key driver of positive feedback as every time someone links to a particular node on a network it makes it that bit more likely someone else will also. This example helps to illustrate the dynamics behind positive feedback and how through these positive feedback loops the system can move or develop in a particular direction very rapidly.
Many real-world networks such as the World Wide Web have proven to have this power law relationship between the size and quantity, where there are just a very few sites with a very large size and very many of a very small size. We should note here again that with the network effect as with all nonlinear systems, things can go both ways. It may have helped to grow the internet to its vast size in a very short period of time which we might cite as a positive thing, but also the network effect is in operation when some negative news about your company goes viral and behind the creation of herd mentalities. Exponentials are such a powerful force for change because the system is not just growing or decreasing, but that due to the feedback loops and synergies within the system and over time there is also another meta-level to the system’s development that is itself increasing this rate of growth or decay to enable very rapid change.