The Simply Complex

Comparing an artificial stock market and the electronic ecosystems, we see the power of complexity theory to address experimentally questions that heretofore science has been powerless to explore, questions like how various "fashions" come and go in a financial market or what kind of organizational structures we might to expect to see on a "second Earth" in the Andromeda galaxy.

The Fingerprints of Complexity

In everyday parlance the term complex is generally used to describe a person or thing that is composed of many interacting components whose behavior and/or structure is just plain hard to understand. The behavior of national economies, the human brain and a rain forest ecosystem are all good examples of complex systems. These examples underscore the point that sometimes a system may be structurally complex, like a mechanical clock, but behavior very simply. In fact, it's the simple, regular behavior of a clock that allows it to serve as a timekeeping device. On the other hand, there are systems, like certain types of a toy rotator, whose structure is very easy to understand but whose behavior is impossible to predict. And, of course, some systems like the brain are complex both in structure and behavior.

As these examples indicate, there's nothing new about complex systems; they've been with us from the time our ancestors crawled up out of the sea. But what is new is that for perhaps the first time in history, we have the knowledge, and more importantly the tools, to study such systems in a controlled, repeatable, scientific fashion. And hopefully this newfound capability will eventually lead to a viable theory of such systems.

Prior to the recent arrival of cheap and powerful computing capabilities, we were hampered in our ability to study complex systems like a national economy or the human immune system because it was simply impractical, too expensive - or too dangerous - to tinker with the system as a whole. Instead, we were limited to biting off bits and pieces of such processes that could be looked at in a laboratory or in some other controlled setting. But with the arrival of today's computers, we can actually build complete silicon surrogates of these systems inside our computing machines, and use these would-be worlds as laboratories within which to look at the workings - and behaviors - of the complex systems of everyday life.

To speak of a system as being complex suggests identifying features separating complex systems from those that are in some sense simple. Here are a few of the most important of these fingerprints of complexity.

Instability

Complex systems tend to have many possible modes of behavior, often shifting between these modes as the result of small changes in some factors governing the system. For instance, the flow of water or oil through a pipe is smooth when the flow velocity is low. But if the velocity is increased beyond a critical level (that depends on the viscosity of the fluid), eddies and whirlpools appear. And if the velocity is increased still further, the frothy, chaotic motion of fully-developed turbulence sets in.

Irreducibility

Complex systems come as a unified whole; they cannot be studied by breaking them into their component parts and looking at the parts in isolation. The behavior of the system is determined by the interaction among the parts, and any tearing of the system into pieces destroys the very aspects that give it ist individual character. A good illustration of this is the well-known problem of protein folding. Every protein is formed as a chain of amino acids, strung together like beads on a necklace. Once the protein has been assembled, it folds up into a unique three-dimensional configuration that determines its function in the living organism, a configuration that is determined completely by the one-dimensionalsequence of amino acids. But in order to know what this final configuration will be, it is simply not possible to separate the question of how the protein will fold into a set of smaller subproblems by cutting the protein at various spots and seeing how these subchains of amino acids fold, and then cementing together somehow the solutions of these individual subproblems. To see how the protein will fold it must be studied as a single, integrated whole.

Adaptability

Complex systems tend to be composed of many intelligent agents, who take decisions and actions on the basis of partial information about the entire system. Moreover, these agents are capable of changing their decision rules on the basis of such information. A driver in a road-traffic network or a trader in a financial market illustrate this point nicely, since in both cases the agent receives partial information about the system he or she is a part of - traffic conditions for the driver, prices and market trends for the trader - and takes actions on the basis of this information. As a result of these actions, both the driver and the trader gain information about what the rest of the system - the other drivers or traders - are doing. The agent can then modify his or her decision rules accordingly. In short, complex systems generally have the capability of learning about their environment and changing their behavioral responses to it in the light of new information.

In passing, let me note that human beings are not the only type of agent that fits this mold. Molecules, corporations and living cells also qualify as intelligent, adaptive agents, who can change their behavior in response to changes in their environment.

Emergence

Complex systems produce surprising behavior; in fact, they produce behavioral patterns and properties that just cannot be predicted from knowledge of their parts taken in isolation. These so-called emergent properties are probably the single most distinguishing feature of complex systems. Everyday tap water illustrates the general idea, as its component parts - hydrogen and oxygen - are both highly flammable gases yet combine to produce a compound that is neither. Thus, the properties of being a liquid and noncombustible are emergent properties arising from the interaction of the hydrogen and oxygen agents.

A similar phenomenon occurs when one considers a collection of independent random quantities, such as the heights of all the people in New York City. Even though the individual numbers in this set are highly variable, the distribution of this set of numbers will form the familiar bell-shaped curve of elementary statistics. This characteristic bell-shaped structure can be thought of as emerging from the interaction of the component elements. Not a single one of the individual heights can correspond to the normal probability distribution, since such a distribution implies a population. Yet when they are all put into interaction by adding and forming their average, the Central Limit Theorem of probability theory tells us that this average and the dispersion around it must obey the bell-shaped distribution.

With these ideas in mind, let's turn to some examples of systems displaying these sorts of features.