 In this module, we will be continuing with our discussion on power laws from a slightly different perspective of probability. One product of this exponential power law feature to nonlinear systems is that they are non-normal, meaning there is statistically no normal average or typical state within the system. To understand this, we need to first talk a bit about what we mean by average and normal. When we say normal, we mean that there is some kind of mean state to the system. In enough samples of the different states within the system, we will be able to compute some average that we can use as a representative of the whole system. So for example, the distribution of people's heights, if we were to plot them, would follow what is called a normal distribution, meaning there will be very many people around the average of say 5 to 6 feet, some a bit larger and some of it smaller than this, say between 3 and 8 feet, but virtually no one outside of this range of states. These normal distributions then have a well-defined centre and then drop off exponentially fast, meaning there is an extraordinary low probability of getting extreme states. Another feature to normal distributions is the so-called law of large numbers, which means that the average of the results obtained from a large number of trials should be close to the expected value and will tend to become closer as more trials are performed. This so-called law is important because it guarantees stable long-term results for the averages of some random events. For example, while a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins. Any winning streak by a player will eventually be overcome by the law of large numbers that will bring the net gains and losses back to an average. The normal distribution holds for many linear systems, physical and chemical, but in the world of non-linearity, the idea that there is such a thing as normal, and we should expect this normal, is no longer applicable. The power law nature to non-linear systems creates what is called a long-tail distribution, meaning extraordinary events that are virtually impossible within normal distributions are possible within non-linear systems. These extraordinary events are referred to as black swans. For example, on October the 19th, 1987, on Black Monday, the Dow Jones Stock Market Index dropped by 22% in a single day. Compared to the typical fluctuations of less than 1%, this was a shift in more than 20 standard deviations from the norm. In a normal distribution, this would be virtually impossible, with something like a 10 to the power of minus 50 chance of it happening. Although in normal distributions, extraordinary events like this are virtually impossible, they are much more common in power law distributions, resulting in a very different graph that has a long tail, sometimes called fat tail. These extraordinary events can then have a dramatic effect on the system's average behavior, essentially rendering this concept of average or normal nonsensical, because if we take a random sample, we will get a certain average, but then if we add just one more node to this, it might be a black swan that will radically alter the average again. So if we go back to our example of measuring people's heights, if we have a room of people and then the tallest person in the world walks in, the average height will only change by a few feet. But say we are measuring people's income, and now the richest person in the world walks in, with an income of many billions, this would so radically alter the average income for it to become nonsensical. In a power law distribution, as we increase the number of samples we take, values will not converge to an average, but will in fact diverge with some exceptions. Asking for an average is like asking how big is a stone, or how long is an average piece of string? Within probability power laws describe an exponential relation between the size of an event and the frequency of it occurring. Many types of systems follow this power law distribution, resulting in a long tail graph, from traffic on the internet to the occurrence of earthquakes and stock market crashes. This power law makes extreme events much more likely and renders our traditional conception of average and normal no longer applicable. Again, this ties back to our main theme of nonlinearity being the product of the nature of interactions between components and over time. Our normal distribution is largely derived from the fact that we are taking random samples from components that have no correlation between them. If I flip a coin now, it will not affect the value that I get when I flip it the next time, and will follow a normal distribution. If you win on one casino table, it will not affect whether I will win on another one and so on. But in these nonlinear systems, things are arranged in a particular way. Large websites are large because of the network effect, and because people have specifically chosen to connect to them. There's nothing random about this. Financial crashes are also similar in nature. In a few lectures from now, we'll be coming back to illustrate how power laws are in fact closely related to fractal geometry.