Complex systems
What do metabolic pathways and ecosystems, the Internet, and propagation of HIV infection have in common? Until a few years ago, the answer would have been very little. The first two examples are biological and shaped by evolution, the third is a human creation, and the fourth is an unwieldy mixture of biology and sociological components. However, in the last few years the answer that has emerged is that they all share similar network architectures. Seemingly out of nowhere, in the span of a few years, network theory has become one of the most visible pieces of the body of knowledge that can be applied to the description, analysis, and understanding of complex systems. New applications are developed at an ever-increasing rate and the promise for future growth is high: Network theory is now an essential ingredient in the study of complex systems. However, before delving into networks themselves it is important to put the overall subject in context, attempt a definition of complexity itself, and present a brief review of the other tools that are used in the analysis of complex systems. The discussion has to start with two important distinctions: First a differentiation between what is complex and what is merely complicated. Second a differentiation between the complexity of the dynamics generated by simple systems and that of complex systems.
Simple systems have a small number of components which act according to well understood laws. Consider what is perhaps the prototypical simple system; the pendulum. The number of parts is small, in fact, one. The system can be described in terms of well-known laws--Newton's equations. The example of the pendulum raises an important point: The need to distinguish between complex systems and complex dynamics: It takes little for a simple system such as the pendulum to generate "complex'' dynamics. A forced pendulum--with gravity being a periodic function of time--is chaotic. In fact one can argue that the driven pendulum contains everything that one needs to know about chaos; the entire dynamical systems textbook by Baker and Gollub is built around this theme. And a pendulum hanging from another pendulum--a double pendulum--is also chaotic.
Complicated systems have a large number of components which have well-defined roles and are governed by well-understood rules. A Boeing 747-400 has, excluding fasteners, three million parts. In complicated systems, such as the Boeing, parts have to work in unison to accomplish a function. One key defect (in one of the many critical parts) brings the entire system to a halt. This is why redundancy is built into the design when system failure is not an option. More importantly, complicated systems have a limited range of responses to environmental changes. Even the most advanced mechanical chronometers can only adjust to a small range of changes in temperature, pressure and humidity before they loose accuracy. And a Boeing without its crew is not able to do much of anything to adjust to something extraordinary.
Complex systems typically have a large number of components which may act according to rules that may change over time and that may not be well understood; the connectivity of the components may be quite plastic and roles may be fluid. Contrast the Boeing 747-400 with a flock of migrating geese. Superficially, the geese are all similar and the flock has far fewer members than the Boeing has parts, so one might be tempted to think that the Boeing is more complex than the flock of geese. However, the flock of migrating geese is an adaptable system, which the Boeing is not. The flock responds to changes in the environment--that is indeed why it migrates--more-over, and unlike what one may naively guess, the migrating geese self-organize without the need for a leader or maestro to tell the rest of the flock what to do. This is clearly revealed by observing the dynamic unrepeated patterns generated by the geese as they adjust their flying formations. Roles in the flock are fluid and one goose at the head of the formation will quickly be replaced by another. This feature of the flock gives it a great deal of robustness as no single goose is essential for the flock's success during the migration. The stock market, a termite colony, cities, or the human brain, are also complex. As for the flock of geese, the number of parts is not the critical issue. The key characteristic is adaptability--the systems respond to external conditions.
It is far from trivial to come up with an all-encompassing definition of complex systems. Nevertheless let us attempt one: A complex system is a system with a large number of elements, building blocks or agents, capable of interacting with each other and with their environment. The interaction between elements may occur only with immediate neighbors or with distant ones; the agents can be all identical or different; they may move in space or occupy fixed positions, and can be in one of two states or of multiple states. The common characteristic of all complex systems is that they display organization without any external organizing principle being applied. The whole is much more that the sum of its parts.
Examples of complex systems are among some of the most elusive and fascinating questions investigated by scientists nowadays: how consciousness arises out of the interactions of the neurons in the brain and between the brain and its environment, how humans create and learn societal rules, or how DNA orchestrates processes in our cells.