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We can
 consider complex systems to be ones formed by a large number of heavily interacting elements. As a result, many of mankind’s greatest challenges come from trying to unravel the behaviour of these systems, such as the climate, the economy, society, the brain, biological development, etc. However, contrary to this, the hydrogen atom, solar system or an ideal gas would be simple systems, despite the fact that in order to study them we need to use in-depth physics concepts and sophisticated mathematics.
However, if everything that is complex is a complex system, what does the new science of complexity bring to the table? Can such wide-ranging systems be tackled with a single perspective? One of the key ideas in complexity studies is that structures appear in these types of systems at all levels, including levels far in excess of those achieved by the interaction between components; as well as this, the structures also show surprising statistical regularities.
At the Complex Systems Group of the Centre for Mathematical Research, we focus on two major lines of research: one, natural disasters and meteorological phenomena, resulting from the complex activity of the Earth’s system, and the other, the structure of the information in human communication, produced by the areas of the brain responsible for this and the relationship between the speakers.
Natural Disasters and Meteorological Phenomena
Many natural phenomena occur in sudden and short intense bursts of activity that interrupt long periods of calm, invalidating any description in terms of continuous functions. Examples include seismology (earthquakes), meteorology(hurricanes, intense rain) and ecology (forest fires). Our study into these complex systems has found simple patterns common to all of them.
The most paradigmatic law in this case is Gutenberg-Richter’s law on earthquakes: the distribution of the magnitude follows an exponential law. However, in terms of energy, this distribution is transformed into a power law, which involves some very specific properties: the lack of characteristic scales and the divergence of mean energy. So, if we visit a country and ask what large earthquakes are like here, the fact of the matter is that there is no answer to this straightforward question.
Phenomena of this kind are difficult to generate, but a critical ramification process may fulfil this role. Let’s consider a tribe where the women have an exact average of one daughter (the number of male children is irrelevant, if there is at least one “alpha male”). This population lies on the limit between extinction and survival, which only occurs when the average number of daughters per mother is exactly one (obviously daughters who reach reproductive age). This appears to be the case with earthquakes: during their evolution, they are always on the limit between attenuation and intensification, which would make them intrinsically unpredictable. These and other related studies will enable us to improve our assessment of seismic risk in different regions
As well as working with earthquakes, our group also studies whether for an apparently very different phenomenon, such as hurricanes, there is a similar law on energy dissipation. However surprising this may seem, and as far as we know, these distributions have not been measured to date. The implications of the results involved in the tricky problem of predicting hurricane intensification would be extremely relevant. And, as if that were not enough, the distribution of energy from hurricanes enables us to assess clearly the response from this phenomenon in light of global warming indicators (e.g. sea surface temperature).
We are undertaking a similar project for the occurrence of rain. The key lies not in looking at the occurrence of rain by days or months (as has traditionally been the case), but considering “episodic” rain events, similar to earthquakes, that may last as little as a minute (according to the data resolution that we have). We hope that these studies enable us to study the correlation between episodes of rain and the episodes of drought which precede and follow them to provide a better characterisation of these processes that are so relevant to society. Also, by measuring the relationship with other atmospheric variables, such as the amount of water vapour, we expect to obtain accurate information about the origins of atmospheric convection, which will be of huge interest for the parameterisation of these meteorological and climate prediction models.
Human Language
It is curious to note that human language verifies a law similar to the law on earthquake energy distribution: this is the so-called Zipf Law. In effect, if we allocate each word in a text a rarity (so that the most common has a rarity of 1, the next a rarity of 2, etc.), the distribution of this rarity follows a power law. This means that no characteristic rarity exists, but that the texts use all the possible scales of rarity. Also, contrary to what we might think, we neither speak nor write freely, but follow the guideline set out by the Zipf Law.
Surprisingly, some authors have explained this fundamental discovery in terms of a totally random process: a monkey dumbly striking a keyboard would generate a “Zipf” text. This would mean all the complexity of human communication not being reflected in the structure of the texts, which would simply be sequences of uncorrelated symbols.
The aim of our group is to find statistical patterns that prove the existence of complexity, both in human and animal language, and to try and establish a connection between the two. Also, the existence of laws which describe the sequential structure of texts will have as their far-from-negligible subproduct the creation of tools for recognising key words and the automatic classification of documents, which is an over-riding necessity in today’s society of information saturation.
(Image courtesy of CIMMS,