Simply Complexity (8 page)

0 1 0 1 0 1 0 1 0 1 . . .

which happens to be the same time-series as the filing example above.

Now let’s imagine that we, as outside observers, turn up at the office or market and are faced with such a time-series of 0’s and 1’s. This time-series tells us what has happened in the recent past. Our job as office observers, or as market speculators, is to work out what will happen next. Just to remind ourselves, the time-series looks like

0 1 0 1 0 1 0 1 0 1 . . .

and hence looks very ordered in time. Even if we don’t know the systematic rule which the intern is using to move the file in the office scenario, the fact that this time-series is so ordered means that we could probably guess the rule ourselves and even make a reasonably accurate prediction about the next outcome – which we would probably all guess would be a ‘0’ in this example. The same idea holds for the scenario of a financial market – in other words, whether the price moves up or down looks to be predictable in our example, even though the actual value of the price might not. You might think we would be pretty lucky to find such an ordered pattern in a financial market – and you would be right, partially. For example it turns out that the dollar-yen exchange rate did, for several years during the 1990s, indeed have such an underlying pattern.

So, for an office with two shelves, it might be reasonably straightforward to make a prediction as long as the intern follows a systematic rule for changing the file’s position. Likewise for a financial market with an up-down price-series, it might be possible to make a prediction of future up-down movements as long as the collection of traders is operating as a crowd, and can hence generate an ordered series of price movements. We return to this point in
chapter 6
.

Now imagine replacing the systematic intern by a careless one, who moves the file around randomly. In other words,

One file, two shelves, and a careless intern:

This is equivalent to saying that he just flips a coin each day in order to determine the file’s new position, with heads representing ‘1’ and tails representing ‘0’. The time-series will now become completely random, with typical sequences such as

0 1 0 0 0 1 1 0 1 0 . . .

which contain no patterns whatsoever. In other words, the time-series looks completely
disordered in time
. If we were outside observers we would now say that the series looks unpredictable. The same holds for a financial market: if the traders are not acting as a crowd, the price-series is much more likely to resemble this random one shown above, as opposed to the ordered one shown earlier.

In our two-shelf example, the systematic intern gave rise to a series of outcomes which are highly ordered in time, and hence a predictable time-series, while the careless one gave rise to a series of outcomes which are highly disordered in time, and hence an unpredictable time-series. This could be turned into a nice neat take-home message about predictability were it not for the phenomenon of Chaos.

It turns out that even a systematic intern can, if he uses a sufficiently complicated rule and if the number of possible arrangements is sufficiently large, produce a time-series which looks highly disordered in time, and hence unpredictable. Strictly speaking, chaotic time-series do have a predictable pattern in them. However, it is so hard to find it that it might as well not be there – and therein lies the problem with trying to predict the dynamics. So let’s see how this works by considering a setup in which the file-moving rule is complicated and where there are many possible arrangements.

One file, many shelves, and a systematic intern:

We consider the case of many shelves, and hence many possible arrangements of the one file. The intern uses the following systematic rule for determining the file’s next position. Since this rule is quite complicated to explain, we will write it as a list of instructions:

Step 1.
Calculate a number
S
which is given by the shelf number on which the file is located divided by the total number of shelves. In other words,
S
is a number between 0 and 1 which expresses how far up the filing cabinet the file is. So
S
= 1 means the file is on the top shelf and
S
= 0 means the file is on the bottom shelf.
S
= 0.5 means the file is half-way up, and would correspond to the file sitting on shelf 50 in a filing cabinet containing 100 shelves for example.
S
= 0.25 means the file is a quarter of the way up, and would correspond to the file sitting on shelf 25 in a filing cabinet containing 100 shelves.

Step 2.
Suppose the file starts off sitting on a given shelf, and that this corresponds to a particular value of
S
which we will refer to as
S
₁
. In order to work out which shelf to move the file to, which
we call
S
₂
, the intern takes
S
₁
and multiplies it first by (1 –
S
₁
) and then by a number
r
. Let’s choose
r
= 4, and
S
₁
= 0.4. This means that (1 –
S
₁
) = 1 – 0.4 = 0.6 and hence the new shelf location
S
₂
is given by

S
₂
= 4 × 0.4 × 0.6 = 0.96

In mathematical terms, the formula which the intern has used – and which by the way is the only formula in this book – is given by

S
₂
=
r
×
S
₁
× (1 –
S
₁
)

Step 3.
The intern now repeats Step 2, but with
S
₁
replaced by
S
₂
and
S
₂
replaced by
S
₃
. In other words, he uses the formula

S
₃
=
r
×
S
₂
× (1 –
S
₂
)

and hence obtains
S
₃
= 4 × 0.96 × (1 – 0.96) which means that
S
₃
= 0.15.

Step 4.
The intern repeats this process over and over again in order to obtain all subsequent shelf locations. In other words, he obtains
S
₄
from
S
₃
, then
S
₅
from
S
₄
and so on.

If you were to follow this set of instructions expecting the number
S
to eventually settle down to some particular value, you would be in for a surprise – it never does. Not only that, but there is no discernible pattern at all. This is because you have created a time-series which is chaotic. In other words, you have uncovered Chaos. Now, maybe you wouldn’t have expected
S
to ever settle down and therefore you feel like I have wasted your time. If this is the case, let me quickly re-surprise you. Go back and repeat the whole process, but instead of using
r
= 4 you now use any value of
r
between 0 and 1. For example, let’s choose
r
= 0.1, and let’s still use
S
₁
= 0.4 as above. The new shelf location
S
₂
is given by

S
₂
= 0.1 × 0.4 × 0.6 = 0.024

and hence the next shelf location
S
₃
is given by

S
₃
= 0.1 × 0.024 × 0.976 = 0.0023

Keep going with this and you will find shelf locations
S
that get closer and closer to zero, i.e. the file moves quickly to the bottom of the filing cabinet, which corresponds to
S
= 0, and stays there. It is as though the file has been attracted to a particular point in the filing cabinet where it then becomes fixed for all time. We have just uncovered a so-called fixed-point attractor of the system’s dynamics. By contrast, the earlier example with
r
= 4 was very strange in that the file didn’t seem to be attracted to any particular shelf – in fact, the value of
S
never repeats itself. The technical term for this type of behavior is, somewhat unsurprisingly, a strange attractor.

Let’s just take a moment to catch our breath and think through the implications. A systematic intern applying the complicated rule that we wrote down with
r
= 4, and with a filing cabinet with many shelves, will produce Chaos. Although the successive locations, i.e. successive
S
values, looked like they occurred randomly, this is only because the rule was so complicated that it produced a very complicated output time-series. There was still method in the madness, in that the systematic intern knew exactly what he was doing. And unlike a careless intern flipping a coin, he would get the same result on a given day no matter how many times he repeated the calculation. So the file would always end up on the same shelf on a given day. In terms of us as outside observers just looking at this output time-series, we could
if we were really clever
deduce the rule that the systematic intern used – and hence make an accurate prediction about the file’s next location – just by observing the shelf positions over many, many days. In other words, a rule exists and it would be up to us to find it. Actually, I am not sure that I would be able to find it in practice, but at least it is possible in principle – and that is good to know. Indeed, this is very much like when you take over someone’s job in an office if they are suddenly away on sick leave – you know there must be some kind of logic to their filing system, but it can really take a lot of effort to finally work it out.

This filing example has also shown us something else – something rather curious, and ultimately very worrying. The same systematic intern, using the same rule but just changing very slightly the value of the number
r
from
r
= 4 to a number between 0 and 1 (
r
= 0.1 in our case), managed to completely change the time-series
of file locations that was produced. Instead of bouncing around all over the place, the file moved quickly to the bottom of the filing cabinet (
S
= 0) and then just stayed there. And this type of behavior is completely the opposite of Chaos. So we have uncovered an important take-home message for understanding the types of behaviors that a Complex System can show. Even if a system has the same setup – in our case, the same systematic intern, the same rule for changing shelf, the same number of files and the same number of shelves – there can be a wide range of outputs, or in other words a wide range of dynamical behaviors. One such example is Chaos, but there are others.

Maybe you are thinking that the consequences for understanding a given Complex System aren’t actually that bad. Maybe things are either chaotic, and hence the time-series of outputs appears disordered in time and therefore essentially random (e.g.
r
= 4), or they are completely ordered in time (e.g.
r
= 0.1). Not quite, unfortunately. It turns out that the road between these two is also complicated – in other words the route to chaos is quite a rocky road. And in our case, we can take ourselves along this route simply by changing the value of the number
r
. Specifically, we can take the system from the regime where the output time-series is ordered in time, with
r
between 0 and 1, to the regime where the output time-series appears to be disordered in time, with
r
= 4, just by changing the value of the number
r
. As we will see, the variety of possible behaviors that we uncover is enormous. And given that our intern-filing scenario is an extremely simple example of what a Complex System can actually do, it follows that any given real-world Complex System might show an equally wide range of such behaviors – and maybe even more. So let’s investigate more carefully this panorama.

With
r
between 0 and 1, the output time-series is extremely ordered in time. No matter where the file starts off, it quickly makes its way down to
S
= 0, and then just stays there. Let’s suppose that the systematic intern now chooses
r
to be bigger than 1, for example
r
= 2. Starting again with 0.4, the sequence of successive shelf locations is:

0.4 0.48 0.5 0.5 0.5 0.5 . . .

So instead of heading toward the bottom of the filing cabinet, the file heads toward the middle and stays there. Now let’s suppose that the systematic intern had chosen a value slightly larger than
r
= 3, such as
r
= 3.2. In this case, the resulting time-series of file locations eventually repeats itself – in particular, the following pattern emerges:

. . . 0.80 0.51 0.80 0.51 0.80 . . .

In technical jargon, the time-series has become periodic and hence repeats itself after every two steps. It therefore has a period equal to two. This is very strange since there is nothing in the rule which the systematic intern uses which would suggest that the file should move between two shelves in such an ordered way. And yet the file just bounces back and forth between these two shelves like the tick-tock of a reliable clock. Amazing – but things get even stranger on our route to chaos as
r
increases toward
r
= 4.

Suppose that the systematic intern chooses a slightly larger value of
r
, such as
r
= 3.5. The resulting time-series suddenly stops repeating itself after every two steps, and instead repeats itself after every four steps. It has a period of four. So the file will move between four shelves in an ordered way, going back to the same shelf every four steps. Yet all the systematic intern did was to change slightly the number
r
in the rule that he was using.