A complex system is a system whose properties are not fully explained by an understanding of its isolated parts and components: even if you understand each isolated part in detail, you may not fully understand the whole system. It can only be understood in terms of its parts and the interactions between them. Since this is a typical feature of unpredictable paradox or emergent properties, these properties are the key properties of complex systems, and they are the reason why these systems are limited in their predictability. What distinguishes a complex system from a merely complicated one is that some behaviors and patterns emerge in complex systems as a result of the patterns of relationship between the elements.
Complex systems are liteally woven out of many parts: the Latin complexus comes from the Greek pleko or plektos, meaning "to plait or twine". It is difficult to give a concise definition of complex systems and complexity in general, because systems are complex if they can not be described in a simple way. Various definitions of complex systems have been used. A special edition of Science about complex systems Science Vol. 284. No. 5411 (1999) highlighted several:
- A complex system is a highly structured system, which shows structure with variations (Goldenfeld and Kadanoff)
- A complex system is one whose evolution is very sensitive to initial conditions or to small perturbations, one in which the number of independent interacting components is large, or one in which there are multiple pathways by which the system can evolve (Whitesides and Ismagilov)
- A complex system is one that, by design, or function, or both, is difficult to understand and verify (Weng, Bhalla and Iyengar)
- A complex system is one in which there are multiple interactions between many different components (D. Rind)
- Complex systems are systems in process that constantly evolve and unfold over time (W. Brian Arthur).
According to Sandra Mitchell and her book "Biological Complexity and Integrative Pluralism", (Cambridge University Press, 2003), the relations and the emerging properties are the most important characteristic:
- "complex systems can be distinguished from simple objects by having multiple parts that stand in nonsimple relations [...] There is structure or order in the way in which the whole is composed of the parts".
Julio M. Ottino gives the following rough definition of Complex Systems in his article about Complex systems 
- "A complex system is a system with a large number of elements, building blocks or agents, capable of exchanging stimuli with one another 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 state or multiple states. The common characteristic of all complex systems is that they display organization without any external organizing principle being applied. In the most elaborate examples, the agents can learn from past history and modify their states accordingly. Adaptability and robustness are often the byproduct. Part of the system may be altered, and the system may still be able to function."
As examples he lists the human brain, the stock market and cities, and divides all kind of complex systems in three main categories and classes:
a) physical and chemical systems b) biological systems, c) social systems and organizations.
Complex systems are the essential elements of science and technology. Science is all about exploration of complex systems, engineering about exploitation of complex systems. In science we would like to understand complex systems (discover simple rules and equations which lead to complex phenomena), in technology and engineering we would like to master them (create a simple user interface for complex devices and conceal complex tools and instruments behind simple interfaces). Since they are difficult to understand, a central problem in science and engineering is the ability to predict and control the behavior of these systems.
Of course every system can be considered as (more or less) complex, only the degree of complexity varies. It is usually not the mere number of components, parts or elements alone which makes a system complex. Any macroscopic material has a huge number of atoms and molecules. It is the overall heterogenity and diversity of the parts and their relations, and most of all the nontrivial, dynamic organization on multiple scales into intricate structures, patterns, networks and hierarchies.
Complex systems lie often at the border between order and disorder. in other words at the edge of chaos: they are neither completely ordered nor completely random. They are made of different constituents. Often each element is individual and unique, like the whole system itself. As Albert-László Barabási says in his article Taming complexity, Nature Physics Vol. 1 (2005) 68-70, "...most complex systems are not made of identical and undistinguishable components, as gases or magnets are — each gene in a cell or individual in a country has its own characteristic behaviour [...] the components obey neither the extreme disorder of gases, in which a molecule can collide with any other molecule, nor the extreme order of magnets, where spins interact only with their immediate neighbours in a nicely periodic lattice. Rather, in complex systems the interactions form exquisite networks, each component being in contact with selected interaction partners."
Hard to understand
Understanding a complex system requires more than simply assembling a list of isolated parts. Even a detailed description how they are wired together is not enough, because usually the parts are not connected in a rigid way with each other. One needs to understand the network of interactions and the functional consequences, to pay attention to multiple levels of analysis - individuals, interactions, and groups -, to distinguish the different types of dynamic interactions, patterns, temporal sequences and timing of events, and finally to learn the response of the system to different stimuli in different environments.
In contrast to many other simple physical systems, the behavior of a complex system often depends crucially on historical events, for instance on frozen accidents. In order to understand many natural complex systems, one has to know the history of the system. The impact of the various historical events varies, some have vanishing impact, some are very strongly amplified. The corresponding process is also known as path dependence and butterfly effect.
In general, complex systems are characterized by intricate braids of causal relations, in other words they are systems where the question of causation is complicated to untangle. This means they are difficult to understand, to describe or to analyze. Complex systems consist of a large number of mutually interacting and interwoven parts, entities or agents.
Hard to predict
Unpredictable paradox or emergent properties are the key properties of complex systems. Paradox means the cause and effect are contrary to one another, emergent means the effect is visible, but the cause remains unclear. In other words, complex systems have intricate causal structures, which are not evident from the knowledge of the constituents alone. Nino Boccara says in his book "Modeling Complex Systems" (2004) that the appearance of emergent properties is the single most distinguishing feature of complex systems. He lists the following three characteristics common to many complex systems:
- They consist of a large number of interacting agents
- They exhibit emergence; that is, a self-organizing collective behavior difficult to anticipate from the knowledge of the agents' behavior
- Their emergent behavior does not result from the existence of a central controller
Courses about "Complex Systems" often begin with the follwing conclusions after the first lecture:
- Complex systems are the rule, not the exception
- Complex systems arise from interactions between agents
Both statements are not true. They are misleading and not correct in general. Complex systems usually do not arise from interactions between agents. The self-organization and emergence of complex patterns by simple rules is the exception - this is what makes the emergence process interesting in the first place. Complexity does not emerge from simplicity for free. Put some autonomous agents together, and you will probably get conflicts, fights and wars for free. Maybe simple spatial patterns like stripes, heaps, grids or simple networks, too. Everything else requires luck or a very long process of evolution or a sophisticated, deliberate design.
And complex system are not the rule. Complex systems are common here on earth - mostly as a result of a four billion year long evolutionary process - but if we take a look from the ISS, we can see that the complexity on earth is the exception in the vast emptiness of space, and not the rule at all. If you consider the entire galaxy or universe, then the systems by which we are surrounded in daily life are very complex systems. Entropy increases everywhere, only in small pockets we can find exceptions. The earth can be regarded as such an exception, like an island in a variety of simple systems.
- in all living organisms which are subject to evolution
- in all evolving complex adaptive systems which have a long historical background
- in software development, esp. where systems have grown over a long period of time
Designed systems can be complex, too, but more often "complicated" is a better description. Complicated systems are like machines, they are designed with a certain purpose, every part as a certain function, and one key defect (in one of the many critical parts) brings the entire system to a halt.
Models and Tools
The basic tools to explore and to study complex systems are
All of these systems consists of a large number of interacting elements arranged in a certain order. In Multi-Agent Systems the elements are usually mobile, in Cellular Automata they form a rigid grid, and in Random Boolean Networks they form a complex network.
A complex system is often described by several theories and multiple models, depending on the particular perspective. Complex systems (esp. complex adaptive systems, multi-agent systems and social systems) are located in an area of tension between the following behavior poles:
- historical vs. regular behavior (exceptional vs. expected events)
- micro vs. macro behavior (low-level vs high-level patterns)
Their behavior depends neither solely on individual events and accidents nor on universal laws. Both sites play an important role, historical accidents (see for example the principles "sensitivity to initial conditions", butterfly effect, frozen accidents, path dependence) and regular laws. Likewise, the behavior of complex systems depends neither solely on individual microscopic actions nor on macroscopic structures, institutions and organizations. Both layers are important (see for example the principles emergence, swarm intelligence, self-organization). The most interesting behavior occurs in the center between the extremes, if microscopic actions have a strong effect on macroscopic behavior and vice versa, or if historical accidents become global patterns.
An ideal theory would combine both aspects, historical and regular behavior, micro and macro behavior by defining universal "laws of history" or "theories of emergence". Unfortunately, both are very rare. Stuart Kauffman writes in his book "At Home in the Universe" on page 299: "We lack a theory of how the elements of our public lives link into webs of elements that act on one another and transform one another. We call these transformations 'history'. Hence with all the accidents of history, one must engage in a renewed debate: Is there a place for law in the historical sciences? Can we find lawlike patterns, cultural, economic, and otherwise?"
This question is quite similar to the question of Leo Nikolayevich Tolstoy in his epic novel "War and Peace": "Only by taking infinitesimally small units for observation (the differential of history, that is, the individual tendencies of men) and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history.", War and Peace, Leo Tolstoy, Book 11, Chapter 1.
Stephen Wolfram says in this article about Complex Systems Theory:
- "Complex systems theory cuts across the boundaries between conventional scientific disciplines. It makes use of ideas, methods and examples from many disparate fields. And its results should be widely applicable to a great variety of scientific and engineering problems."
Nino Boccara, Modeling Complex Systems, Springer, 2004, ISBN 0-387-40462-7
Portals and Links
The Yahoo! Directory and Google Directory offer many interesting links. Note that the modern theory of Complex Systems was classified as mathematical application from the DMOZ editors of the "Open Directory Project", which is also the base of the Google Directory.
Relevant theories are cybernetics, systems theory, complexity theory, and chaos theory.
- Engineering Complex Systems, Julio Ottino, Nature Vol. 427 (2004) 339