This article largely discusses complex systems as a subject of mathematics and the attempts to emulate physical complex systems with emergent properties. For other scientific and professional disciplines addressing complexity in their fields see the complex systems article and references.

A complex system is a system composed of interconnected parts that as a whole exhibit one or more properties (behavior among the possible properties) not obvious from the properties of the individual parts.^[1] A system’s complexity may be of one of two forms: disorganized complexity and organized complexity.^[2] In essence, disorganized complexity is a matter of a very large number of parts, and organized complexity is a matter of the subject system (quite possibly with only a limited number of parts) exhibiting emergent properties.

Examples of complex systems for which complexity models have been developed include ant colonies, human economies and social structures, climate, nervous systems, cells and living things, including human beings, as well as modern energy or telecommunication infrastructures. Indeed, many systems of interest to humans are complex systems. Complex systems are studied by many areas of natural science, mathematics, and social science. Fields that specialize in the interdisciplinary study of complex systems include systems theory, complexity theory, systems ecology, and cybernetics.

Overview[link]

A complex system is a network of heterogeneous components that interact nonlinearly, to give rise to emergent behavior.^[3] The term complex systems has multiple meanings depending on its scope:

• A specific kind of systems which are complex
• A field of science studying these systems; see further complex systems
• A paradigm that complex systems have to be studied with non-linear dynamics; see further complexity

Various informal descriptions of complex systems have been put forward, and these may give some insight into their properties. A special edition of Science about complex systems ^[4] highlighted several of these:

• A complex system is a highly structured system, which shows structure with variations (N. 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).

History[link]

Although one can argue that humans have been studying complex systems for thousands of years, the modern scientific study of complex systems is relatively young when compared to conventional science areas with simple system assumption such as physics and chemistry. The history of the scientific study of these systems follows several different research trends.

In the area of mathematics, arguably the largest contribution to the study of complex systems was the discovery of chaos in deterministic systems, a feature of certain dynamical systems that is strongly related to nonlinearity.^[5] The study of neural networks was also integral in advancing the mathematics needed to study complex systems.

The notion of self-organizing systems is tied up to work in nonequilibrium thermodynamics, including that pioneered by chemist and Nobel laureate Ilya Prigogine in his study of dissipative structures.

Types of complex systems[link]

Chaotic systems[link]

For a dynamical system to be classified as chaotic, it must have the following properties:^[6]

Assign z to z² minus the conjugate of z, plus the original value of the pixel for each pixel, then count how many cycles it took when the absolute value of z exceeds two; inversion (borders are inner set), so that you can see that it threatens to fail that third condition, even if it meets condition two.

1. it must be sensitive to initial conditions,
2. it must be topologically mixing, and
3. its periodic orbits must be dense.

Sensitivity to initial conditions means that each point in such a system is arbitrarily closely approximated by other points with significantly different future trajectories. Thus, an arbitrarily small perturbation of the current trajectory may lead to significantly different future behavior.

Complex adaptive systems[link]

Complex adaptive systems (CAS) are special cases of complex systems. They are complex in that they are diverse and made up of multiple interconnected elements and adaptive in that they have the capacity to change and learn from experience. Examples of complex adaptive systems include the stock market, social insect and ant colonies, the biosphere and the ecosystem, the brain and the immune system, the cell and the developing embryo, manufacturing businesses and any human social group-based endeavor in a cultural and social system such as political parties or communities. This includes some large-scale online systems, such as collaborative tagging or social bookmarking systems.

Nonlinear system[link]

The behavior of nonlinear systems is not subject to the principle of superposition while that of linear systems is subject to superposition. Thus, a nonlinear system is one whose behavior can't be expressed as a sum of the behaviors of its parts (or of their multiples).

Topics on complex systems[link]

Features of complex systems[link]

Complex systems may have the following features:

Cascading Failures

Due to the strong coupling between components in complex systems, a failure in one or more components can lead to cascading failures which may have catastrophic consequences on the functioning of the system.^[7]

Difficult to determine boundaries

It can be difficult to determine the boundaries of a complex system. The decision is ultimately made by the observer.

Complex systems may be open

Complex systems are usually open systems — that is, they exist in a thermodynamic gradient and dissipate energy. In other words, complex systems are frequently far from energetic equilibrium: but despite this flux, there may be pattern stability, see synergetics.

Complex systems may have a memory

The history of a complex system may be important. Because complex systems are dynamical systems they change over time, and prior states may have an influence on present states. More formally, complex systems often exhibit hysteresis.

Complex systems may be nested

The components of a complex system may themselves be complex systems. For example, an economy is made up of organisations, which are made up of people, which are made up of cells - all of which are complex systems.

Dynamic network of multiplicity

As well as coupling rules, the dynamic network of a complex system is important. Small-world or scale-free networks^[8][9][10] which have many local interactions and a smaller number of inter-area connections are often employed. Natural complex systems often exhibit such topologies. In the human cortex for example, we see dense local connectivity and a few very long axon projections between regions inside the cortex and to other brain regions.

May produce emergent phenomena

Complex systems may exhibit behaviors that are emergent, which is to say that while the results may be sufficiently determined by the activity of the systems' basic constituents, they may have properties that can only be studied at a higher level. For example, the termites in a mound have physiology, biochemistry and biological development that are at one level of analysis, but their social behavior and mound building is a property that emerges from the collection of termites and needs to be analysed at a different level.

Relationships are non-linear

In practical terms, this means a small perturbation may cause a large effect (see butterfly effect), a proportional effect, or even no effect at all. In linear systems, effect is always directly proportional to cause. See nonlinearity.

Relationships contain feedback loops

Both negative (damping) and positive (amplifying) feedback are always found in complex systems. The effects of an element's behaviour are fed back to in such a way that the element itself is altered.

See also[link]

Systems science portal

• Agent-based model
• Biological organisation
• Complex (disambiguation)
• Complexity (disambiguation)
• Complex adaptive system
• Complex network
• Dissipative system
• Fractals
• Innovation butterfly
• Mixed Reality
• System equivalence
• Systems theory

References[link]

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (November 2007)

Further reading[link]

• Chu, Dominique (2011). Complexity: Against Systems. Theory in Biosciences, Springer Verlag. [1]

• Rocha, Luis M. (1999). "Complex Systems Modeling: Using Metaphors From Nature in Simulation and Scientific Models". BITS: Computer and Communications News. Computing, Information, and Communications Division. Los Alamos National Laboratory. November 1999

• Ignazio Licata & Ammar Sakaji (eds) (2008). Physics of Emergence and Organization, ISBN 978-981-277-994-6, World Scientific and Imperial College Press.

• James S. Kim, Hyper Emotional Society, Version 9. Knol. 2009 Nov 25.

• Alfred Hübler, Cory Stephenson, Dave Lyon, Ryan Swindeman (2011). Fabrication and programming of large physically evolving networks Complexity, 16(5), pp. 7–8

• De Toni, Alberto and Comello, Luca (2011). Journey into Complexity. Udine: Lulu. ISBN 978-1-4452-6078-5. 

External links[link]

Articles/General Information

Look up complex system in Wiktionary, the free dictionary.

• Introduction to complex systems-short course by Shlomo havlin
• Complex systems in scholarpedia.
• (European) Complex Systems Society
• (Australian) Complex systems research network.
• Complex Systems Modeling based on Luis M. Rocha, 1999.
• CRM Complex systems research group
• The Center for Complex Systems Research, Univ. of Illinois at Urbana-Champaign
• FuturICT - Exploring and Managing our Future