Most learned people are aware that there is some sort of a connection between financial markets and biological evolution, between earthquakes and stock market crashes. They know it all has something to do with fractals and chaos, and that such connections are the subject of the relatively new realm of science called “complexity theory.”
But the general understanding rarely seems to go much deeper. Few truly grasp the common thread tying together all these various phenomena, and scientists themselves clearly have trouble explaining the thread in terms that don’t sound vague or wishy-washy. There was a time when complexity theory actually was somewhat wishy-washy; when scientists saw the similarities and patterns between various fields of study but were at a loss as to how or why they arose.
But that time has passed. At this point there is an explanation available that grants a huge amount of clarity about the common processes of the natural world. And unlike the complex systems it explains, the theory isn’t all that complicated. Systems as disparate as world economies, rivers, forest fires, earthquakes, the human brain, and even the internet all display a behavior known as self-organized criticality (SOC). Understand SOC, and you will understand a great deal about the relationship between all these different types of systems.
The canonical example of a system that displays SOC is a pile of sand. (The three scientists who “discovered” SOC, Bak, Tang, and Wiesenfeld, used a sandpile model to present their theory to the world in the late 1980s, and it has remained the most straightforward presentation.) You may have noticed, or it may seem right when you think about it, that when you pile up sand at the beach, the slope of the cone of sand you make is always the same no matter how big or small the cone is. Gather up some sand, move your hands away, and the sandpile spontaneously falls into a very regular shape, always with the same slope.
Well it just so happens that this is the “critical slope” of the pile. Sandpiles, like many other naturally-occurring aggregates of a vast number of individual units, are attracted to their so-called “critical point”. Without fine-tuning or careful arrangement, they just fall into that state. The critical point is where complex systems like sandpileswant to be.
At the critical point, sandpiles behave in a very peculiar way. If you perturb a pile by dropping a single grain of sand anywhere onto it, literally anything could happen. The grain could either slide down the pile a short distance and stop, it could knock a few grains down with it, or it could cause a huge avalanche and cave in the entire side of the sandpile.
In other words a sand avalanche of any scale is possible, and at the critical point, the severity of an avalanche exactly correlates with how likely it is to occur. Explicitly, the added grain of sand is 10 times more likely to knock 10 grains down the side of the pile than it is to displace 100 grains, and a 100-grain avalanche is in turn 10 times more likely than an avalanche involving 1000 grains. Small avalanches occur more often than big ones, but any size avalanche can and will happen if you spend long enough dropping grains onto the pile.
Slightly more technically speaking, the size of an avalanche is inversely proportional to its frequency. If you dropped grains of sand onto the pile over and over for days and recorded the size of each of the resulting avalanches, then graphed the results, the size-frequency proportionality would emerge. This correlation is known variously as 1/f noise, power law or fractal behavior, and scale-invariance. It occurs because, at the critical point, an infinity appears in the equation governing the behavior of the system, rendering such behavior unknowable. (Those interested in the math of critical points should read more here.)
SOC behavior is exhibited all over nature, wherever small perturbations (like the addition of single grains of sand) happen to large systems (like sandpiles). For example, small vibrations of tectonic plates can cause earthquakes of any size, with a severe earthquake being much less likely than a small one. The price drop of a single stock can have little or no effect on the stock market as a whole, or it can spur a chain of events that leads to a major stock market crash and economic depression. The extinction of a single biological species can bring down five others with it, or five hundred. The firing of a neuron in the brain can die down without effect, or it can cascade and grow into a conscious thought. A military skirmish can lead to a couple of others, or world war. In all these cases, “avalanches” of any size are possible, and with self-organized critical systems there’s really no predicting what size avalanche will occur as a result of a given perturbation.
To use one more example, I am “perturbing” the internet by posting this article. I can be confident that this post is more likely to generate a few hundred hits than a few thousand, but who knows. With a chain reaction of Facebook shares and Tweets, anything is possible: thus the allure of blogging!
The brain, the internet, sandpiles, tectonic plates, the weather, stock markets, ecosystems, and literally countless other systems exhibit SOC, but there is as yet no general theory as to what exactly constitutes a system which causes it to self-organize around its critical point. This question occupies many minds and blackboards around the world.
My final observation on the matter of self-organized criticality concerns its relevance. Despite the fact that it explains so much about the way the world works, SOC isn’t part of the general lexicon. People don’t talk about it. Apart from complexity theorists, even most scientists don’t talk about it. This is because scientists are in the business of predicting events in the world around us, and scientific theories have always been valued according to how well the predictions they make match reality. SOC turns this completely on its head. SOC is a theory about the impossibility of prediction. Not only can’t SOC tell you how severe the next San Francisco earthquake is going to be, it is telling you that the question isn’t answerable. That the earthquake could be any size. That we might as well stop trying to guess.
This kind of message is hard to swallow if you’re judging science by its old standards. Self-organized criticality, and complexity theory as a whole, is certainly a new kind of science. It is more relevant than any theory before it as an explanation of how nature works, but not because it provides a means of determining what is going to happen in the future. It can’t predict the next event that will occur in a complex system. Instead it predicts the pattern of events spanning the past, present, and future all at once, in no order.