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Complex Systems

What are complex systems? There is no single answer, but there are two partly alternative approaches to the problem, and both have to do with the mathematical-computational model used to study a system: we can speak of the complexity of the algorithm used to study the system, or of the cognitive complexity needed to understand the problem.

At the heart of all these investigations lies the concept of a model. When we study a phenomenon, even if we approach it experimentally—for example, a disease spreading in a bacterial culture—we always try to eliminate all unnecessary elements: does the disease affect only a certain type of bacteria or is it generic? Does it depend, and how, on light, temperature, or the acidity of the medium? What happens if the culture is grown on a solid medium or instead in a liquid one? Does it spread only among neighboring cells or among all of them?

Once we have a clearer picture of the situation, we begin to use mathematical/computational formulations in order to design more sophisticated experiments, or even experiments that would be impossible to carry out in the physical world.

For example, if we cannot determine whether there is an incubation phase of the disease, we can ask ourselves: what happens if I assume that bacteria have only two states, “healthy” and “sick”, and that a sick bacterium can infect only its neighbors for a certain time, after which it dies? With this schematization, the model can easily be implemented on a computer, allowing me to rapidly perform thousands of “virtual experiments” and obtain measurements that I can compare with experimental ones. A sophisticated statistical analysis is then needed to validate the model and to discard, or evaluate, alternative models.

Clearly, no model this simplified will be able to reproduce the experimental phenomenology exactly, and certainly, once the general picture is understood, details will have to be added that will inevitably require greater computing power to be evaluated. But it is clear that the basic model provides a “basis” for many variations, and therefore it is worth studying it in detail.

The models obtained are generally composed of a certain number of relationships, which often take the “mathematical form” of equations. In general, these equations describe what happens to the system after a short time interval if we know its current state. The problem is often to predict the behavior of the system over long times or when we impose particular conditions. Moreover, in many cases, the number of equations grows with the number of elements in the system—for example, the number of bacteria in a culture or the number of atoms in a sample—and therefore we also ask whether predictions made using a large number of elements will differ from those, easier to carry out, made with only a few elements.

The simplest case is the one in which each element behaves almost independently and reacts in a linear, that is, proportional, way to variations in the initial conditions or to those otherwise imposed. For example, the model of an ideal gas, made up of little balls that do not interact except when they collide, with elastic collisions, is rather simple. We can easily derive conservation laws—for the number of particles and for energy—and also, with some approximation, derive collective laws, such as the equation of state of ideal gases. Many of the models studied at school and university are of this type.

The great advantage of these models is that we can predict what they will do without following their evolution step by step. For example, the equation of state of the ideal gas allows us to predict the increase in pressure when the temperature is raised or the volume is reduced, without concerning ourselves with the intermediate states.

Obviously, such simple models do not explain, fortunately, all of reality. To stay with the gas example, the little-ball model does not explain why a gas cools when it expands freely, nor can it be used to reproduce the phenomena of liquefaction and evaporation.

However, we can introduce additional elements. A real gas is made up of molecules, not little balls. What happens if I use composite elements? And then real atoms and molecules also interact at a distance, not only during collisions. Which element allows us to explain real phenomena?

In the case of phase transitions, liquefaction and evaporation, the fundamental element is long-range interactions, which obviously complicates the study. In the history of scientific research, many intellectual resources have been devoted to studying the effect of these “additions”.

But today, since we have computers at our disposal, we can use them to explore the consequences of these equations, in a very general sense, which we are unable to analyze with pencil and paper.

A first example is deterministic chaos. In this case we have equations, sometimes even just one, that present no computational difficulty in being simulated, but that exhibit behaviors that are difficult to predict. A classic “surprising” effect is sensitivity to changes in initial conditions. In linear models—that is, simple ones; think of throwing a ball—the effect of a variation in the initial conditions is a proportional variation in the final situation. In fact, this is how we tackle many everyday tasks: we try to perform an action, and if the result is not the desired one, we adjust, for example, the force of the throw or the initial direction. But if the system is highly nonlinear, as in many amusement-park games—or, more simply, as in the behavior of a ball containing a movable weight—it becomes very difficult to understand what to do in order to achieve the goal.

In other cases, unpredictability arises when we add elements that, in themselves, may even behave simply. The most historically relevant case is the three-body problem. Newton had managed to derive Kepler’s laws from the law of universal gravitation, a fairly complicated but ultimately not complex result. However, the same laws, applied to the problem of more than one planet and the Sun, give rise to trajectories that can be very complicated, and chaotic.

Thus, complexity from the cognitive point of view lies in the ability to “predict” its evolution, which is also connected to the possibility of describing the state of the system using few symbols, because our brain has a finite, and small, processing capacity.

Conversely, from the point of view of computers, models that are “complex” for humans—cognitive complexity—often do not present particular computational difficulties. But being able to “simulate” a system is not equivalent, for us, to “understanding” it. For example, for us it is more important, when studying the basic model, to know that the system changes its behavior—for example, becomes chaotic or starts to boil—as some parameter varies, rather than to reproduce its behavior in detail.

However, there are mathematical problems that are “difficult” even for computers, and these are, for example, those underlying cryptography. In this case we speak of algorithmic complexity.

So, to sum things up somewhat, let us look at some examples of complex systems that are the object of theoretical study by researchers at the center; see also the research section.

• Chaotic systems, in which problems such as the divergence of trajectories, changes in behavior, bifurcations, as parameters vary, or, conversely, the efforts needed to control or synchronize them, are studied.
• Systems composed of many interacting parts, in which sudden variations as parameters change—phase transitions, critical phenomena—are studied, both at equilibrium and out of equilibrium.
• Systems in which the constituent elements interact through a network structure, in which case the interesting problems are those induced by changes in topology or in the intensity of connections.
• Active systems with cooperative and/or competitive interactions, such as evolutionary or economic systems, in which there is no “configuration” capable of satisfying all needs.
• Disordered systems, again in which it is not easy to find the condition that minimizes the number of unsatisfied constraints.

The numerical analysis of such problems obviously has strong connections with computation and statistics. Moreover, problems with a direct application of these principles can be found in physics, chemistry, mathematics, and computer science.

But we do not limit ourselves to theoretical study; the classes described above are found in many applied problems.

A typical example is our atmosphere and its meteorological manifestations: it is largely made up of gas molecules and water vapor interacting through simple collisions, but on a large scale, precisely that of atmospheric phenomena, it displays extremely complex and difficult-to-predict behavior.

Another example is a network for the production, distribution, and use of electricity, consisting of a large number of nodes and connections linking fundamental units governed by relatively simple and known laws of operation: the global behavior of the system, for example the spread of a failure, can depend critically on the topology of the network and on the dynamics of its components.

Another example is that of amorphous materials, glasses and ceramics, and granular media, sand, rice, corn: here too, the laws of microscopic interaction that govern the dynamics of the individual elementary constituents give rise to peculiar macroscopic behaviors. The systems under examination can at times be assimilated to solids and to fluids, even though they are not exactly either true solids or true fluids.

Continuing with the examples, one can mention the cell cycle and the mechanisms of gene regulation related to it: even when we are able to reconstruct with good approximation the map of interactions among the genes that regulate their expression, we remain, in any case, far from understanding what makes “the whole” tendentially stable with respect to external perturbations, some of which, however, are capable of producing violent variations up to the loss of coherence of the entire structure.

A prime example of a complex system is the brain. It consists of an enormous number of elementary constituents, the neurons, which exchange simple electrical impulses with one another through a network of synaptic connections that shows no apparent regularity in the number of connections between pairs of neurons. And yet this system is globally capable of perceiving sensations, processing them, and, even more extraordinarily, conceptualizing them.

Clearly, at the immediately higher level we have the cognitive sciences, and psychology, with their applications.

But evolving biological systems too, from viruses to bacteria to ecosystems, are complex, because each species—or rather, each individual—must find a way to maximize the probability of survival and reproduction at the expense of others, and this, paradoxically, can also lead to cooperative phenomena.

These themes also have a direct application in medicine.

By combining the biological and cognitive, or rather cybernetic, domains, we can introduce strategies, that is, behaviors that do not depend only on immediate interactions, but that may have goals and “remember” past interactions. This can happen even without the need for a nervous system, for example in plants or bacteria.

Clearly, there are deep connections with sociology and economics or finance, which are examples of complex systems made up of complicated agents, but in which it can sometimes be shown that collective behavior is in fact the result of very simple strategies.

Human behaviors are also of primary importance for environmental sustainability, with themes that concern both infrastructure protection and urban planning choices.

Finally, in conclusion, it should be remembered that information science and engineering, computer science, touches on all these aspects to some extent, both because computers are everywhere and because in many cases they connect human beings, with their typical behaviors.

Lastly, the ever-increasing role that artificial intelligences play in all these fields should not be overlooked, in addition to being a subject of study in its own right.