Simply Complexity (5 page)

Figure 2.1
The number of possible arrangements for a pile containing (top) two files labelled A and B and (bottom) three files labelled A, B and C

Two files, one shelf, and one careless intern:

Suppose the files are labelled A and B. As demonstrated in
figure 2.1
, there are only two possible arrangements for this two-file pile:

Arrangement 1: file B on top of file A

Arrangement 2: file A on top of file B

For this particular case of only two files, both these arrangements are essentially ordered. In other words, if your careless intern accidentally rearranges your files, the worst he can do is just to reverse the order. If you find the files are not in the order you wanted, just turn the pile upside down.

Three files, one shelf, and one careless intern:

Now let’s imagine that your job is slightly busier, and that you now have three files in the pile, rather than two. Let’s suppose these files are labelled A, B and C. Here comes the bad news: even though we have only increased the number of files by 50 percent, there are now
three times as many
possible arrangements. These are shown in
figure 2.1
, and are listed below:

Arrangement 1: file C on top of file B, on top of file A

Arrangement 2: file B on top of file C, on top of file A

Arrangement 3: file A on top of file C, on top of file B

Arrangement 4: file C on top of file A, on top of file B

Arrangement 5: file B on top of file A, on top of file C

Arrangement 6: file A on top of file B, on top of file C

So going from just two files to three files means that we have gone from two possible arrangements to six. So now your careless intern can accidentally rearrange the pile in
six
ways.

How many arrangements would there be for four, five, or more files? Obviously more, but how many more? It turns out that there is a simple way of working this number out. Suppose we are building a pile out of three files. There is a choice of three possible files to place at the bottom (A, B or C). Pick one – for example, A. This leaves two possible files in the middle. Again pick one – for example, C. Then there is only one file left to go on top – file B. In other words, we have three possibilities at the bottom, multiplied by two in the middle, and one on top. Writing this in terms of math, this gives a total number of possibilities as:

(3 at the bottom) × (2 in the middle) × (1 on top) = 3 × 2 × 1 = 6, as shown in
figure 2.1
.

More than three files, one shelf, and one careless intern:

For four files – labelled A, B, C and D – we therefore have 4 × 3 × 2 × 1 = 24 possible arrangements. For five files we will have 5 × 4 × 3 × 2 × 1 = 120 possible arrangements. But all this sounds like a pretty unrealistic office. After all, having ten files in a pile isn’t uncommon. So let’s have a look at what would happen for ten files. Following the same idea as above, you can see that the number of possible arrangements will be 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 which turns out to be more than three-and-a-half million. And that really is bad news since it means that your careless intern can accidentally rearrange the pile in more than three-and-a-half million ways!

This gives us our first take-home message about collections of objects. The number of things that can happen to a collection of objects – and in particular the number of arrangements of these objects – quickly becomes very large as you increase the number of objects.

Now, imagine that it is the day before you are going on vacation, and your boss hands you a pile of ten files. She tells you that she has spent the whole day organizing the pile into a special
order, according to her own priorities. She also tells you that you should start work on them, beginning at the top, as soon as you get back from vacation. So you place them on your desk, and give everyone strict instructions to leave them alone. Off you go on vacation, and duly empty your mind, aided by the occasional cocktail. Once back in the office, you find a note that your intern has left you – “Very sorry, I accidentally knocked over your files while you were away. But I put them back in a neat pile for you, so no harm done!” No harm done? There are more than three-and-a-half-million possible arrangements, and you have forgotten the particular arrangement which your boss carefully prepared before you left on vacation. Now imagine that you try to randomly rearrange them, hoping that the special arrangement will magically appear before your eyes. Here comes the really bad news. Suppose you spend ten seconds on every arrangement – that means you could search six arrangements per minute, and therefore 360 arrangements per hour. But since the exact number of arrangements is actually 3,628,800, it will take you approximately 10,000 hours to search all of them – and 10,000 hours means that it will take you more than one year even if you don’t stop to sleep, eat or go to the bathroom. So unless you have a very patient boss, you will likely be out of a job before you find the correct arrangement.

In the above story, we imagined that your intern accidentally pushed the whole pile over and hence instantly took the pile from maximum order to maximum disorder. In many other situations, things will move between order and disorder in a more gentle way. Imagine that instead of the whole pile being knocked over all in one go, your intern just randomly changed the position of one file each day. Going back to our picture with only three files, you can see that even after only a couple of days the new pile is likely to be quite different from the original one. Admittedly, the more files there are in the pile, the longer this process of order-to-disorder will take – but in the end, disorder rules.

So, our office filing story tells us that it is quite easy to disorder something which is ordered, while it requires a long time and a
lot
of care to reorder something that is disordered. Exactly how long either of these takes will depend on what is doing the disordering or reordering. But one thing is for sure:
there is a natural tendency for something that is ordered to become disordered as time goes by. In contrast, something that is disordered is highly unlikely to order itself without any additional help
. And herein lies our interest in order and disorder. We have already established that a Complex System such as the traffic or a financial market, can spontaneously move from order to disorder and back again. At the same time, we know that a Complex System contains a collection of objects in a similar way to a pile of files. So how is it that a Complex System can move from order to disorder and back again, all by itself, while something simple like a pile of files cannot?

There must be some magic ingredient that a Complex System has – but a pile of files or a bag of socks does not have – and which therefore enables the Complex System in question to create order out of thin air all by itself. To help us understand the nature of this magic ingredient, we need to do an experiment:

Grab a ruler from your desk. Now try to balance it upright on your desk. Impossible. Now try to balance it on your outstretched open hand. Again impossible . . .
unless you move your hand continually to counteract the wobbling ruler.

It turns out that this ruler problem is very similar to the file problem that we discussed earlier. The files can easily go from an ordered to a disordered state, just like a momentarily upright ruler on a desk can easily fall over. But to reorder the files, or to help the ruler stay upright, would require a helping hand. In the case of the files, it requires a very kind boss who is willing to come in and redo all the hard work of prioritization. In the case of the ruler, it requires the actions of a skillful handler to help the ruler stay upright.

The examples of the files and the ruler give us the key to understanding why a Complex System is much more complicated
than a collection of objects such as files or socks. In other words, these examples give us the clue to help explain exactly what makes a Complex System so complex, as opposed to merely being a bit complicated. So let’s think more deeply about these examples: the only reason that you can balance the ruler on your hand as opposed to the desk, is because your eye notices the movements of the ruler, and then feeds this information to your brain which then
feeds back
the information to your hand in the form of a movement. The same holds for the reordering of the pile of files: it would need your boss to
feed back
the original information into the pile that she injected about the various files’ priorities. And that is the answer – feedback – which is a term which appeared on our list of key ingredients of a Complex System in
chapter 1
.

As we will see, feedback can arise in a given system in a variety of different ways. It can be built into the objects themselves – for example, humans have a memory of the past which can affect their decisions in the present. Or it can be information or influence fed into the system from the outside, as in the case of the balancing stick, or the announcement of news in a market. In the case of traffic, it can be the information that a driver obtains from looking at the cars around him, or listening to traffic reports on the radio.

It doesn’t matter where it comes from, it is still feedback – and it is feedback which enables order to “kick in” in different ways and at different times. Feedback can create order in a disordered pile of files, and can order a tumbling ruler into its upright position. However it is typically very hard to see such feedback operate at the level of the individual objects in a particular Complex System – hence it can appear to an outside observer that the order appears out of thin air. This is particularly true if the feedback takes the form of information, since information is not a tangible object. Since drivers in traffic and traders in markets continually input and output information about their own and others’ actions, we can begin to see why traffic jams and market crashes could possibly appear out of thin air without any apparent cause. The magic ingredient is the feedback of information.

In terms of our order/disorder story, the upright ruler is an ordered state while the ruler tumbling to the ground is a disordered one. As long as you can keep the ruler upright, you have managed to maintain it in an ordered state. However, you cannot keep this up forever. It takes concentration, and that makes you hungry – and you eventually need to eat. In other words, the origin of the order for the upright ruler is feedback, and this feedback requires the input of energy.

Now, let’s take this a step further. The energy which we use to create the feedback for the ruler, comes from the food we eat. And all the food we eat can be traced back to plants. This is even true for meat and dairy products – they come from animals who themselves ate plants. So it all comes down to plants – and plants get their energy from that great energy source in the sky: the Sun. In other words: the Sun represents the root cause of the pockets of order that we observe around us. This is actually quite a deep statement, since it means that the Sun is what helps us buck the general trend from order to disorder. This is even true when we build buildings or create other ordered structures using machines and materials such as concrete. Machines are made of metal and run on gasoline: and gasoline, metal and concrete all originate from the natural resources found on Earth – and these in turn owe their existence to the Solar System and hence the Sun.

So the Sun is the root cause of the pockets of order that we see all around us. And these pockets of order are not restricted to inanimate objects such as files or upright rulers. The town where you live is an example of a pocket of order. It contains many people, organized into houses and streets – and all within the single pocket defined by the town’s boundaries. Going further, each of us humans is individually a pocket of molecules which all happen to be piled up into a particular region of space, i.e. within the confines of our own particular body.

We have seen that collections of objects such as a pile of files will,
in the absence of any feedback
, tend to become increasingly disordered. Unfortunately, it turns out that the same is true of the Universe as a whole, and everything in it – including us.

Let me explain the background to this horrifying news. All the evidence gathered so far by scientists suggests that the Universe is isolated. It doesn’t touch anything and nothing touches it. Most importantly, there is no feedback of any kind from other Universes – hence there is no “invisible hand” to help keep it ordered. In technical jargon, the Universe is a closed system – and unfortunately there is a fundamental law of physics which states that:
The amount of disorder in a closed system increases as time goes by.
So can we use this law as an excuse for an untidy office? Yes and no. Based on what we saw with the files, we can certainly see why it would make sense in an office setting. However this law is only strictly true for closed systems – and truly closed systems are very rare. In fact, the Universe is the only truly closed system that we know of.