By Greg Fisher
Recently I got in to a very interesting discussion, which led me to articulate to my interlocutor (and myself!) the difference between Chaos Theory and Complexity Theory. I thought I’d write this down in the form of a blog article. I should stress that you should only read further if such technical differences blow your skirt up. If you have a life, you should probably read something else.
Chaos Theory is the study of fascinating yet deterministic systems, whereas complex systems are not deterministic. At first blush, if we were to contrast the words chaotic and complex as used in every day language, we might be forgiven for believing that Chaos Theory is the antithesis of determinism. But it is not.
By determinism I am referring to the idea of a clockwork universe, made famous by Laplace, in which all of the rules of the universe are fixed. In this type of universe, as Laplace pointed out, if we knew enough information about the current state of the universe in addition to all of its fundamental and unchanging laws, we would be able both to calculate the entire history of the universe and to predict its entire future. There would be no room for free will, which would be seen merely as an illusion.
[Note to reader: I would draw your attention to Tony Smith's reply to the first version of this blog below, on the subject of "computational irreducible". My reference to the word "determinism" here should read "classical determinism": computational irreducibility does mean that deterministic systems can also be inherently uncertain; however the broad thrust of this article still stands, that there are fundamental differences between chaotic and complex systems.]
Chaos theory studies these mechanistic types of systems but it tends to emphasise the principle of feedback whereby two variables are influenced by each other: this can lead to non-linearity and the variables behaving in seemingly chaotic ways. Hence the name, Chaos Theory.
An important insight of Chaos Theory is the sensitivity of a chaotic system to initial conditions due to the non-linearity of the system. What this means is that if the initial conditions of a chaotic system were changed microscopically, then over a long enough period of time the outcome of the whole system will be completely different. This is often referred to as The Butterfly Effect. However, it is important to emphasise that if the initial conditions of the chaotic system were unchanged between two simulations to an infinite degree of precision, the outcome of the two will be the same over any period of time. So the butterfly effect really only serves to contrast the outcomes in two marginally different systems that are still deterministic i.e. machine-like. In one simulation, the butterfly flapped its wings, in the other it did not.
It is also worth mentioning that the butterfly effect can be abused when, for example, a person says “a butterfly flapping its wings in Wigan causes a hurricane in Huddersfield”. This is a classical Newtonian interpretation of the butterfly effect, as if the hurricane happened because of the butterfly. It may be that in a chaotic system, the simulation whereby the butterfly metaphorically didn’t flat its wings was the simulation in which there was a hurricane. In which case, naughty butterfly. The point is that the butterfly effect is meant to illustrate the sensitivity to initial conditions, not causality.
In complex systems, there is a concept known as a global cascade, which is similar to what people often mean by the butterfly effect but it is in fact fundamentally different. A global cascade is basically a network-wide domino effect that occurs in a dynamic network, made famous by Duncan Watts in 2002. Watts showed that sometimes a complex system proved robust in the face of a modest shock (it might just wobble slightly); but in other instances, the same shock might cascade across the system, showing it to be fragile.
Dynamic networks – or complex systems – are very different to the systems studied in Chaos Theory. They contain a number of constituent parts (“agents”) that interact with and adapt to each other over time. Perhaps the most important feature of complex systems, which is a key differentiator from chaotic systems, is the concept of emergence. Emergence “breaks” the idea of determinism because it means the outcome of some interaction can be inherently unpredictable. In large systems, macro features often emerge that cannot be traced back to any particular event or agent.
To understand the concept of emergence further, we can look at water. We know the qualities of oxygen atoms and we know the qualities of hydrogen atoms, so presumably we can determine the qualities of H20, water? Actually we cannot. Really, however freakishly this might sound, we cannot. It turns out that we are familiar with the qualities of water only because we have observed them empirically. The properties of water are emergent. If any reader wished to delve further in to this, I would recommend Stuart Kauffman’s book Re-Inventing the Sacred. In this book, Kauffman wrote that
“it is something of a quiet scandal that physicists have largely given up trying to reason ‘upward’ from the ultimate physical laws to larger-scale events in the universe”
The reason for the inability to reason ‘upwards’ from hydrogen and oxygen to water is because the properties of water are emergent and therefore indeterminable from the properties of the constituent atoms.
A useful way of distinguishing between chaotic and complex systems is to illustrate how uncertainty arises in each type of system. In chaotic systems, uncertainty is due to the practical inability to know the initial conditions of a system. If the initial conditions were either ‘x’ or ‘a tiny (immeasurable) variation on x’, we know that when non-linearity kicks in, this slight variation will eventually make a large difference in the outcome. But in complex systems, uncertainty is inherent in the system because of the concept of emergence: even if we could measure the initial conditions today to an infinite degree of precision (for the serious geeks: I am ignoring Heisenberg’s uncertainty principle here), we still cannot determine the future. However, as I noted in my article Patterns Amid Complexity, this does not mean the future is random because patterns are an important – and often repeating – feature of complex systems.
As a final note, the term “edge of chaos” gets bandied around a lot and, confusingly, this term is normally associated with complex systems. It represents a point that sits along a spectrum with determinism at the one end and randomness (not chaos) at the other. The edge of chaos is where you have enough structure / patterns in the system that it is not random but also enough fluidity and emergent creativity that it’s not deterministic.
To conclude, it is tempting to believe that Chaos is a highly complex type of complex system. But it isn’t – Chaos theorists have revealed some excellent insights, like sensitivity to initial conditions, but Chaos Theory is still a study of deterministic systems. To understand non-deterministic systems, like social systems, it’s necessary to look at complex systems and Complexity theory.