Complexity? It's All Relative.

 

Complexity is a measure of the total amount of structured information (which is measured in bits) that is contained within a system and reflects many of its fundamental properties, such as:

• Potential -  the ability to evolve, survive
• Functionality - the set of distinct functions the system is able to perform
• Robustness - the ability to function correctly in the presence of endogenous/exogenous uncertainties

In biology, the above can be combined in one single property known as fitness.

Like any mathematically sound metric our complexity metric is bounded (metrics that can attain infinite values are generally not so useful). The upper bound, which is of great interest, is called critical complexity and tells us how far the system can go with its current structure.

Because of the existence of critical complexity, complexity itself is a relative measure. This means that all statements, such as, "this system is very complex, that one is not", are without value until you refer complexity to its corresponding bounds. Each system in the Universe has its own complexity bounds, in addition to its current value.  Because of this a small company can, in effect, be relatively more complex than a large one, precisely because it operates closer to its own complexity limit.  Let us see a hypothetical example.

Imagine two companies: one is very large, the other small. Suppose each one operates in a multi-storey building and that each one is hiring new employees. Imagine also that the small company has reached the limit in terms of office space while the larger company is constantly adding new floors. This is illustrated in the figure below.

In this hypothetical situation, the smaller company has reached its maximum capacity and adding new employees will only make things worse. It is critically complex and, with its current structure, it cannot grow - it has reached its physiological growth limit and can do two things:

• "Add more floors" (this is equivalent to increasing its critical complexity - one way to achieve this is via acquisitions or mergers)
• Restructure the business

If a growing business doesn't increase its own critical complexity at the appropriate rate it will reach a situation of "saturation". If you "add floors" at a rate that is not high enough, the business will become progressively less resilient and will ultimately reach a situation in which it will not be able to function properly, not to mention facing extreme events.

Complexity is a disease of our modern times (more or less like high cholesterol, which is often consequence of our lifestyles). Globalisation, technology, or uncertainty in the economy are making life complex and it is increasing the complexity of businesses themselves. An apparently healthy business may hide (but not for long!) very high complexity. Just like very high cholesterol levels are rarely a good omen, the same may be said of high complexity. This is why companies should run a complexity health-check on a regular basis.

So, the next time you hear someone say that something is complex, ask them about critical complexity. It's all relative!

 

 

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Complexity: A Link Between Science and Art?

Serious science starts when you begin to measure. According to this philosophy we constantly apply our complexity technology  in attempts to measure entities/phenomena/situations that so far haven't been quantified in rigorous scientific terms. Of course we can always apply our subjective perceptions of the reality that surrounds us so as to classify and rank, for example, beauty, fear, risk, sophistication, stress, elegance, pleasure, anger, workload, etc., etc. Based on our perceptions we make decisions, we select strategies, we make investments. When it comes to actually measuring certain perceptions complexity may be a very useful proxy.

Let's consider, for example, art. Let's suppose that we wish to measure the amount of pleasure resulting from the contemplation of a work of art, say a painting. We can postulate the following conjecture: the pleasure one receives when contemplating a work of art is proportional to its complexity. This is of course a simple assumption but it will suffice to illustrate the main concept of this short note. Modern art produces often paintings which consist of a few lines or splashes on a canvas. You just walk past. When, instead, you stand in front of a painting by, say, Rembrandt van Rijn, you experience awe and admiration. Now why would that be case? Evidently, painting something of the calibre of The Night Watch is not matter of taking a spray gun and producing something with the aid of previous ingested chemical substances. Modern "art" versus a masterpiece. Minutes of delirium versus years of hard work. Splashes versus intricate details. Clear, but how do you actually compare them?

We have measured the complexity of ten paintings by Leonardo da Vinci and Rembrandt. The results are reported below without further comments.

Leonardo

and Rembrandt

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Why is Resilience (in economics) such a difficult concept to grasp?

Resilience, put in layman's terms, is the capacity to withstand shocks, or impacts. For an engineers it a very useful characteristic of materials, just like Young's modulus, the Poisson ratio of the coefficient of thermal expansion. But high resilience doesn't necessarily mean high performance, or vice versa. Take carbon fibres, for example. They can have Young's modulus of 700 Gigapacals (GPa)  and a tensile strength of 20 GPa while steel, for example, has Young's modulus of 200 GPa and a tensile strength of 1-2 Gpa. And yet, carbon fibers (as well as alloys with a high carbon content) are very fragile while steel is, in general, ductile. Basically, carbon fibres have fantastic performance in terms of stiffness and strength but responds very poorly to impacts and shocks.

What has all this got to do with economics? Our economy is extremely turbulent (and this is only just the beginning!) and chaotic, which means that it is dominated by shocks and, sometime, by extreme events (like the unexpected failure of a huge bank or corporation, or default of a country which needs to be bailed out, like Ireland, Greece, Portugal, or natural events such as tsunamis). Such extreme events send out shock waves into the global economy which, in virtue of its interconnectedness, propagates them very quickly. This can cause problems to numerous businesses even on the other side of the globe. Basically, the economy is a super-huge dynamic and densely interconnected network in which the nodes are corporations, banks, countries and even single individuals (depending on the level of detail we are willing to go to). It so happens that today, very frequently, bad things happen at the nodes of this network. The network is in a state of permanent fibrillation. It appears that the intensity of this fibrillation will increase, as will the number of extreme events. Basically, our global economy will become more and more turbulent. By the way, we use the word 'turbulence' with nonchalance but it is an extremely complex phenomenon in fluid dynamics with very involved mathematics behind it - luckily, people somehow get it! And that's good. What is not so good is that people don't get the concept of resilience. And resilience is a very important concept not just in engineering but also in economics. This is because in turbulence it is high resilience that may mean the difference between survival and collapse. High resilience can in fact be seen as s sort of stability. It is not necessary to have high performance to be resilient (or stable). In general, these two attributes of a system are independent. To explain this difficult (some say it is counter-intuitive) concept, let us consider Formula 1 cars: extreme performance, for very short periods of time, extreme sensitivity to small defects with, often, extreme consequences. Sometimes, it is better to sacrifice performance and gain resilience but this is not always possible. In Formula 1 there is no place for compromise. Winning is the only thing that counts.

But let's get back to resilience versus performance and try to reinforce the fact that the two are independent. Suppose a doctor analyzes blood and concentrates on the levels of cholesterol and, say, glucose. You can have the following combinations (this is of course a highly simplified picture):

Cholesterol: high, glucose: low
Cholesterol: low, glucose:high
Cholesterol: low, glucose: low
Cholesterol: high, glucose: high

You don't need to have high cholesterol to have high glucose concentration. And you don't need to have low glucose levels to have low levels of cholesterol.

Considering, say, the economy of a country, we can have the following conditions:

Performance: high, resilience: low
Performance: low, resilience:high
Performance: low, resilience: low
Performance: high, resilience: high

Just because the German economy performs better than that of many countries it doesn't mean it is also more resilient. This is certainly not intuitive but there are many examples in which simplistic linear thinking and intuition fail. Where were all the experts just before the sub-prime bubble exploded?




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Measuring the Complexity of Fractals

We don't need to convince anyone of the importance of fractals to science. The question we wish to address is measuring the complexity of fractals. When it comes to more traditional shapes, geometries or structures such as buildings, plants, works of art or even music, it is fairly easy to rank them according to what we perceive as intricacy or complexity. But when it comes to fractals the situation is a bit different. Fractals contain elaborate structures of immense depth and dimensionality that is not easy to grasp by simple visual inspection or intuition.

We have used OntoNet to measure the complexity of the two fractals illustrated above. Which one is the most complex of the two? And by how much? The answer is the following:

Fractal on the left - complexity = 968.8

Fractal on the right - complexity = 172.9

This means that the first fractal is about 5.6 times more complex. At first sight this may not be obvious as the image on the right appears to be more intricate with much more local detail. However, the image on the left presents more global structure hence it is more complex. The other image is more scattered with smaller local details and globally speaking it is less complex . This means that it  transmits less structured information, which is precisely what complexity quantifies. Finally, below we illustrate the complexity map of the fractal on the left hand side.

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Is it progress if a cannibal uses knife and fork?

The North American XB-70 Valkyrie was a (beautiful!) tri-sonic bomber developed in the early sixties. It was, even by today's standards, an exceptionally sophisticated and advanced machine. What strikes is the short time it took to develop and build - we intentionally omit the information here since, by today's standards it would make many (aerospace) engineers blush. Why is it  that in the days of slide-rules they could beat today's supercomputers and all other technological goodies? A few reasons are:

• One company did everything - there was no useless geographical dispersion, management was simpler.
• Complex systems have been successfully built even though complexity was not a design goal - the reason is that product development and manufacturing was much less complex.
• Engineers were better trained than today - they understood mechanics better that today's youngsters.
• The average age of engineers was much higher than today.
• Companies had clearer roadmaps and were more motivated - today it's all about shareholders value not about building great planes.
• Aerospace companies were run by people who understood the business.
• Designers didn't have to struggle with super-complex super-huge software systems.
• Companies were profitable (because they were run by people who understood the business) hence were not forced to squeeze every penny out of sub-contractors, causing them to deliver worse results ....
• Because of the above, companies could do plenty of R&D - today, R&D is where (incompetent) management make the first cost cuts.

Today, highly complex products are engineered without taking complexity into account. When coupled with extremely complex and dispersed multi-cultural manufacturing, assembly, procurement, design and management issues you run into trouble if you don't keep complexity under control. It is not surprising that TODAY people doubt that man ever went to the Moon! In fact, those who make similar claims cannot possibly conceive of such a complex project being viable because of their poor preparation.

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Do We Really Understand Nature?

According to the Millennium Project the biggest global challenges facing humanity are those illustrated in the image above. The image conveys a holistic message which some of us already appreciate: everything is connected with everything else. The economy isn't indicated explicitly in the above image but, evidently, it's there, just as the industry, commerce, finance, religions, etc. Indeed a very complex scenario. The point is not to list everything but to merely point out that we live in a highly interconnected and dynamic world. We of course agree with the above picture.

As we have repeatedly pointed out in our previous articles, under similar circumstances:

• it is impossible to make predictions - in fact, even the current economic crisis (of planetary proportions) has not been forecast
• only very rough estimates can be attempted
• there is no such thing as precision
• it is impossible to isolate "cause-effect" statements as everything is linked
• optimization is unjustified - one should seek acceptable solutions, not pursue perfection

The well known Principle of Incompatibility states in fact that "high precision is incompatible with high complexity". However, this fundamental principle, which applies to all facets of human existence, as well as in Nature, goes unnoticed. Neglecting the Principle of Incompatibility constitutes a tacit and embarrassing admission of ignorance. One such example is that of ratings. While the concept of rating lies at the very heart of our economy, and, from a point of view of principle, it is a necessary concept and tool, something is terribly wrong. A rating, as we know, measures the Probability of Default (PoD). Ratings are stratified according to classes. One example of such classes are shown below:

Class       PoD

1              =<0.05%

2              0.05% - 0.1%

3              0.1% - 0.2%

4              0.2% - 0.4%

5              0.4% - 0.7%

6              0.7% - 1.0%

etc.

A rating affects the way stocks of a given company are traded - this is precisely its function. What is shocking in the above numbers, however, is the precision. A PoD of 0.11% puts a company in class 3, while a 0.99 in class 2. How can this be so? Isn't the world  supposed to be a highly complex system? Clearly, if even a crisis of planetary proportions cannot be forecast, it not only points to high complexity (see the Principle of Incompatibility) but it also says a lot about all the Business Intelligence technology that is used in economics, finance, or management and decision making. So, where does all this precision in ratings come from? From a parallel virtual universe of equations and numbers in which everything is possible but which, unfortunately, does not map well onto reality. But the understanding of the real universe cannot be based on a parallel virtual universe which is incorrect.

The above example of PoD stratification reflects very little understanding of Nature and of its mechanisms. In fact, economic crises of global proportions suddenly happen. As Aristotle wrote in his Nikomachean Ethics: an educated mind is distinguished by the fact that it is content with that degree of accuracy which the nature of things permits, and by the fact that it does not seek exactness where only approximation is possible.

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Driving Complexity-To-Target: Application to Portfolio Design

Driving the complexity of a given system to a prescribed target value has numerous applications, ranging from engineering (who wouldn't want a simpler design that performs according to specs?) to management, advanced portfolio design, wealth management or investment strategy.

But more than just complexity it is also the robustness of systems that is of most concern. When considering portfolios both diversification and volatility are of concern
We know that in system design (and this applies to portfolios) the mini-max principle, whereby you maximise something (e.g. the expected return) while minimising at the same time something else (e.g. risk) leads to inherently fragile solutions. Taking simultaneously many things to the limit is of course possible but the price one pays is a rigid and fragile solution: you basically push yourself into a very tight corner of  the design space where you have little margin of manoeuvre in case things go wrong. And things do go wrong. Especially if you think that most things in life are linear and follow a Gaussian distribution you should prepare yourself for a handful of surprises.

Portfolio diversification and design can be accomplished differently based on complexity and, in particular, on these two simple facts:

• High complexity increases exposure - a less complex portfolio is better than a more complex one.
• A less complex portfolio accomplishes better diversification (more or along the lines of the MPT and Markowitz logic).

Let us see an example. Suppose you want to build a portfolio based on the Dow Jones Industrial Average Index and its components. Without going into unnecessary technicalities, below is an example of our first portfolio. We observe that:

Its complexity is 64.3 (pretty close to the critical value of 68.75)
Entropy is 823
Robustness is 66.8%
Rating: 2 stars

Nothing to celebrate.

Suppose now that you wish to increase the robustness to, say, 85%. Using our Complexity-To-Target Technology it is possible to "force" the robustness of the portfolio to this target value. Since robustness and complexity are linked it is possible to do this either for robustness or complexity or even both. The new portfolio is illustrated below.

Complexity is now 50.9
Entropy is 542 - this tells us that the behaviour of the portfolio is substantially more predictable
Robustness is 84.9%
Rating: 4 stars

The hubs of the portfolio (red discs) have now changed but that is another matter.



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