Anything that can be called theory – a theory is something that can be verified empirically in a laboratory – usually has some characteristic constant, a metric, an equation, or some theorems. Think of the theory of gravity, or the theory of relativity. “Theory” means “Explanation for an observable fact, which has been extensively tested, fits all the known observations and can be used to predict how the fact will behave”.

We have trouble with the so-called “theory of complex systems”. There are a few complexity centers around the world involved in the study of “complex systems”, but none of these has ever produced a metric of complexity that would respect, for example the Second Law of Thermodynamics, or that would be bounded – one fundamental requisite for metrics that are rooted in physics and not in video games.

But that is not all. “Complexity science” hasn’t even produced a definition of complexity. All you will ever hear is that “a complex system is one that has spontaneously emerged from the interaction of a number of agents” or “a system that is composed of many agents and that its behaviour cannot be deduced from studying the behaviour of a single component.” Or you can also run into the more romantic version – “a complex system is one in which the whole is greater than the sum of the parts”. As far as I know, all systems in the Universe do all these things. Just think of:

- Stars form galaxies
- Water molecules form waves
- Starlings form (very beautiful) storms
- People form societies
- Motor cars (you cannot deduce what a car does by studying the crankshaft)
- Computers

Also, a good definition is one that hints a metric. In our view, complexity is not a property, reflecting spontaneous structure formation without external choreography – that’s what physics is about – it is a quantifiable attribute of all systems, natural or manmade. Some systems have a lot of complexity some don’t. Some systems have a lot of mass, or energy, others don’t. But we’ve already said all this, and on numerous occasions. So, where’s the theory?

The goal of this short blog is to illustrate the first one of a series of theorems that form part of our Quantitative Complexity Theory, or the QCT. This is the first such theorem, which speaks of the complexity of coupled systems.

An example of a coupled system is illustrated in the figure below, where the nomenclature of the above theorem is used.

If there are no coupling terms, i.e. alpha is 0, then the complexity of the whole is equal to the sum of the complexities of A and B. When there exist coupling terms, complexity can only increase.

However, it must be said that the above theorem is a particular case of a more general one and that the above proof is based on a very simple entry-wise matrix norm, with p=1. In this case, the theorem confirms that indeed the whole is always greater than the sum of the parts, providing there is interaction between the components of a system. If the components don’t interact, you can forget the romantic part and just sum up the complexities of the components.

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