In information theory, complexity is the number of properties transmitted by an object and detected by an observer. Imagine that you, as any manager, try to predict the expenses of your company. You have some contracts with different providers. There are some expenses as taxes that must be payed every year; you are paying some invoices monthly as the cost of electricity and telephone, and you can be paying the raw materials every three months. In a similar way this can be done with incomes.
In this example your cost function is the integration of three different cost frequencies. In order to make the problem clearer, I am going to consider that the payments are continuous following a sine curve, instead of a set of discrete payments, but the reasoning continues being valid.
In that situation, our benefit function should be an aspect similar to this one:
Graph 1 Generated Periodic Benefit Function in the domains of time and frequency
As we can see the signal at the frequency domain has three frequency components. This function has been generated by a computer then it can be easily replied. In a perfect situation we can recover the full signal, from a part of it, and we could make precise forecasting about the requirements of credit at any time.
Following the initial definition of complexity, this company would have an amount of complexity of three, we need to know only the amplitude of three frequencies in order to know precisely the full information of the system.
Following different approaches to complexity, a company like that would not be complex. A complex system has a certain degree of uncertainty. When the behavior of a system can be predicted exactly, we consider that the system is complicated instead of complex.
The previous signal can be got from a set of points at the time domain. An electrical engineer named Nyquist demonstrated that a signal like that (band limited) can be perfectly recovered if the signal is sampled at the double of the highest frequency. This is the dream of any stock exchange investor. Unfortunately, the stock exchange has not a behavior like that. The stock exchange is really complex instead of complicated.
A complex system has not only a welldefined structure; it contains certain amount of uncertainty. In the previous example, we can know the mean of our phone bills but some months we receive different values. That is the reason why, management is more related to complexity management than perfect forecasting.
Graph 2 Complex systems of different degree of uncertainty (1, 5, 6 and 10) in both time and frequency domains.
A very robust economy could have the behavior of the first plot at the previous graph. For instance, we do not know exactly the cost of the electricity or the phone bill but it has a little variability. In that situation we cannot recover perfectly the behavior of system (the signal is not band limited) but as the main frequencies are very clear we could get a good approximation sampling the signal with a frequency higher than Nyquist’s.
This case can be the case of a little startup business. Initially there are not incomes and the expenses are mainly fixes. The company can have losses but their costs can be very predictable as the main bills are the rent of an office, electricity and phone, and interest of loans. It is very easy to manage. There is not a direct relationship between benefit and manageability.
With the second plot we can see that the behavior of the system is preserved at the frequency domain but it is more difficult to identify the system at the time domain. We could know nothing about the behavior of the system in the futures instants with precision, but we can have a better knowledge about the mean behavior along time using a low pass filter.
This does not seem interesting in order to manage a little business, but analyzing the low frequency of the shares of a company at the stock exchange, we could get better throughput of the trading. This can sound odd, but it is well known that the prices of oil increase in winter and decrease in summer because the demand is periodic. Many companies can be affected by different periodic events with different frequencies. Mathematical models for trading are based on similar suppositions.
Looking at the last plot you can see that in a very uncertain situation we can know nothing about the behavior of the system. That is the reason why mathematical economic models cannot work in a turbulent environment. In that case, investors usually search for a portfolio with lower volatility.
In the case that you are a manager of a complex company, the uncertainty is inside you own company and you must make actions in order to reduce uncertainty. There are many actions that can be done to do it, for instance, contracting insurances or flat tariffs. Although the benefit can be lower, you will be able to control the company better under unexpected events.
In order to finish, I would like to point out that as the frequency domain plots shows, uncertainty and structure are similar. The differences are that structure is characterized and uncertainty is not. In a real case, functions are not totally band limited, real structures have an infinite set of harmonics. Increasing structure, we are increasing the energy at the higher frequencies as we do when the uncertainty (characterized here as white noise) increases.
Considering complexity as the energy provided by the addition of structure and uncertainty has a clear physical sense. Thinking in the band limited signals and Nyquist’s theorem, robustness can be understood as the differences between the energy of low frequency and the energy of high frequency that can make our system more predictable through sampling (the measures that the controllers get in order to make managing decisions).
On the other hand, this concept of robustness is showing us that uncertainty is not better or worse than structure. Uncertainty can be better than a structure with the same amount of energy because the distribution of the energy at the frequency domain is flat (considering white noise), the component of high frequency is limited and the energy is distributed between low and high frequencies, however high frequency structure does not provide value if the value of frequency is higher than the sampling one. For the manager is not signal yet, it is only noise.
Although this discussion is good in order to analyze in theory some points, a frequency analysis of an economy can become useless in a real case and it is better to use other techniques in order to measure complexity, in the same way as it is better to use the Discrete Cosine Transform in order to compress TV signals than Fourier’s Transform.
Mr. Luis Díaz Saco Executive President saconsulting advanced consultancy services

Nowadays, he is Executive President of Saconsulting Advanced Consultancy Services. He has been Head of Corporate Technology Innovation at Soluziona Quality and Environment and R & D & I Project Auditor of AENOR. He has acted as Innovation Advisor of Endesa Red representing that company in Workgroups of the Smartgrids Technology Platform at Brussels 