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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 [[Emergence|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_vs_Complex|complicated]] one is that some behaviors and patterns emerge in complex systems as a result of the patterns of relationship between the elements. 

==Definitions==

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|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 (journal)|Science]]'' about '''complex systems''' [http://www.sciencemag.org/content/vol284/issue5411/ 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 [[Butterfly Effect|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).

Julio M. Ottino gives the following rough definition of Complex Systems in his article about Complex systems
[http://www.chem-biol-eng.northwestern.edu/FacultyPDF_Papers/Complex%20Systems%20Perspective%20JMO.pdf]
: "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 [http://en.wikipedia.org/wiki/Science science] we would like to understand complex systems (discover simple rules and equations which lead to complex phenomena), in [http://en.wikipedia.org/wiki/Technology technology] and engineering we would like to master them (create a simple [http://en.wikipedia.org/wiki/User_interface 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.

== Properties ==

=== Ingredients ===

Of course every system can be considered as (more or less) complex, only 
the degree of [[Complexity|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|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 [http://www.nature.com/nphys/journal/v1/n2/pdf/nphys162.pdf 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 Accident|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|path dependence]] and
[[Butterfly Effect|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 [[Emergence|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 [[Agent|agents]]
# They exhibit [[Emergence|emergence]]; that is, a [[Self-Organization|self-organizing]] collective behavior difficult to anticipate from the knowledge of the agents' behavior
# Their [[Emergence|emergent]] behavior does not result from the existence of a central controller

== Misunderstandings ==

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|self-organization]] and [[Emergence|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|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.

On the earth itself one can find complex systems
everywhere - especially at places where life is.
[[Complexity]] can be found everywhere where [[Evolution|evolution]]
is at work,

* 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 vs Complex|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

* [[Agent-Based Model]]s and [[Multi-Agent System]]s
* [[Cellular Automata]]
* [[Random Boolean Network]]s

All of these systems consists of a large number of interacting
elements arranged in a certain order. In [[Multi-Agent System]]s 
the elements are usually mobile, in [[Cellular Automata]] they
form a rigid grid, and in [[Random Boolean Network]]s they
form a [[Complex Network|complex network]].

== Theories == 

A complex system is often described by several theories and multiple models, depending 
on the particular perspective.
Complex systems (esp. [[Complex_Adaptive_System|complex adaptive systems]], 
[[Multi-Agent_System|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|butterfly effect]], 
[[Frozen Accident|frozen accidents]], [[Path Dependence|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|emergence]], [[Swarm Intelligence|swarm intelligence]], 
[[Self-Organization|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.

== Books ==

Nino Boccara, ''Modeling Complex Systems'', Springer, 2004, ISBN 0-387-40462-7

== Portals and Links ==

The [http://dir.yahoo.com/Science/Complex_Systems/ Yahoo! Directory] and [http://directory.google.com/Top/Science/Math/Applications/Complex_Systems/ Google Directory] offer many interesting links.
Note that the modern theory of [http://dmoz.org/Science/Math/Applications/Complex_Systems/ 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|systems theory]], complexity theory, and chaos theory.

* [http://www.chem-biol-eng.northwestern.edu/Nature_Concept.pdf Engineering Complex Systems], Julio Ottino, Nature Vol. 427 (2004) 339


[[Category:Systems]]