 Part of our definition for linear systems is that the relationship between input and output is related in a linear fashion. The ratio of input to output might be two times as much, ten times as much, or even a thousand times. It's not important. This is merely a qualitative difference. What is important is that this ratio between input and output is itself not increasing or decreasing. But as we have seen, when we have 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 phenomena is captured within mathematical notation as an exponential. The exponential symbol describes how we take a variable and we times it by another, not just once, but we in fact iterate on this process, meaning we take that output and we feed it back into computing the next output. Thus the amount we are increasing by each time itself increases. So let's take a quick example of exponential growth so as to give us an intuition for it. Say I want 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 in to affect the population at the next second, like a snowball rolling down a hill. It will take a number of hours before our test tube is just three percent full. But within the next five seconds, as it approaches noon, it will increase to one hundred percent of the test 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. We'll be discussing further the implications of this type of growth later on when we get into the dynamics and nonlinear systems. But for the moment, 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 the other. 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 side lengths of A will have a volume of A cubed, 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 two times the side length. Another example from biology is the nonlinear scaling within the metabolic rate versus size of mammals. The metabolic rate is basically how much energy one needs per day to stay alive, and it scales relative to the mammals 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 relationship between components within a system that is the key source of this nonlinearity. The so-called Meklav's law comes from the world of IT, and it derives from the simple observation. Every time we add a new computer to a network, we have the possibility of adding as many more links as there are computers already in the network. As each new person who joins the network makes it more valuable for everyone else, Meklav's law leads to the value or power of a network increasing in proportion to the square of the number of nodes in the network. This is, of course, not just restricted to computer networks, but is a feature of all networks, and thus it is given the more general name of 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 that 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 size and quantity, where there are just a very few sites with a very large size, and very many with 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. The key takeaway from this section on exponentials is to get a sense of the qualitatively different nature of growth within linear and nonlinear systems. As exponential growth means that the system is not just growing or decreasing, but that due to the positive 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.