A complex adaptive system is a system in which a perfect understanding of the individual parts does not automatically convey a perfect understanding of the whole system's behavior.^[1] The study of complex adaptive systems, a subset of nonlinear dynamical systems,^[2] is highly interdisciplinary and blends insights from the natural and social sciences to develop system-level models and insights that allow for heterogeneous agents, phase transition, and emergent behavior.^[3]

They are complex in that they are dynamic networks of interactions, and their relationships are not aggregations of the individual static entities, i.e., the behavior of the ensemble is not predicted by the behavior of the components. They are adaptive in that the individual and collective behavior mutate and self-organize corresponding to the change-initiating micro-event or collection of events.^[4][5][1] They are a "complex macroscopic collection" of relatively "similar and partially connected micro-structures" formed in order to adapt to the changing environment and increase their survivability as a macro-structure.^[4][5][6]

The term complex adaptive systems, or complexity science, is often used to describe the loosely organized academic field that has grown up around the study of such systems. Complexity science is not a single theory—it encompasses more than one theoretical framework and is highly interdisciplinary, seeking the answers to some fundamental questions about living, adaptable, changeable systems. The study of CAS focuses on complex, emergent and macroscopic properties of the system.^[6][7][8] John H. Holland said that CAS "are systems that have a large numbers of components, often called agents, that interact and adapt or learn."^[9]

Typical examples of complex adaptive systems include: climate; cities; firms; markets; governments; industries; ecosystems; social networks; power grids; animal swarms; traffic flows; social insect (e.g. ant) colonies;^[10] the brain and the immune system; and the cell and the developing embryo. Human social group-based endeavors, such as political parties, communities, geopolitical organizations, war, and terrorist networks are also considered CAS.^[10][11][12] The internet and cyberspace—composed, collaborated, and managed by a complex mix of human–computer interactions, is also regarded as a complex adaptive system.^[13][14][15] CAS can be hierarchical, but more often exhibit aspects of "self-organization."^[16]

What distinguishes a CAS from a pure multi-agent system (MAS) is the focus on top-level properties and features like self-similarity, complexity, emergence and self-organization. A MAS is defined as a system composed of multiple interacting agents; whereas in CAS, the agents as well as the system are adaptive and the system is self-similar. A CAS is a complex, self-similar collectivity of interacting, adaptive agents. Complex Adaptive Systems are characterized by a high degree of adaptive capacity, giving them resilience in the face of perturbation.

Other important properties are adaptation (or homeostasis), communication, cooperation, specialization, spatial and temporal organization, and reproduction. They can be found on all levels: cells specialize, adapt and reproduce themselves just like larger organisms do. Communication and cooperation take place on all levels, from the agent to the system level. The forces driving co-operation between agents in such a system, in some cases, can be analyzed with game theory.

Some of the most important characteristics of complex systems are:^[17]

• The number of elements is sufficiently large that conventional descriptions (e.g. a system of differential equations) are not only impractical, but cease to assist in understanding the system. Moreover, the elements interact dynamically, and the interactions can be physical or involve the exchange of information
• Such interactions are rich, i.e. any element or sub-system in the system is affected by and affects several other elements or sub-systems
• The interactions are non-linear: small changes in inputs, physical interactions or stimuli can cause large effects or very significant changes in outputs
• Interactions are primarily but not exclusively with immediate neighbours and the nature of the influence is modulated
• Any interaction can feed back onto itself directly or after a number of intervening stages. Such feedback can vary in quality. This is known as recurrency
• The overall behavior of the system of elements is not predicted by the behavior of the individual elements
• Such systems may be open and it may be difficult or impossible to define system boundaries
• Complex systems operate under far from equilibrium conditions. There has to be a constant flow of energy to maintain the organization of the system
• Complex systems have a history. They evolve and their past is co-responsible for their present behaviour
• Elements in the system may be ignorant of the behaviour of the system as a whole, responding only to the information or physical stimuli available to them locally

Robert Axelrod & Michael D. Cohen^[18] identify a series of key terms from a modeling perspective:

• Strategy, a conditional action pattern that indicates what to do in which circumstances
• Artifact, a material resource that has definite location and can respond to the action of agents
• Agent, a collection of properties, strategies & capabilities for interacting with artifacts & other agents
• Population, a collection of agents, or, in some situations, collections of strategies
• System, a larger collection, including one or more populations of agents and possibly also artifacts
• Type, all the agents (or strategies) in a population that have some characteristic in common
• Variety, the diversity of types within a population or system
• Interaction pattern, the recurring regularities of contact among types within a system
• Space (physical), location in geographical space & time of agents and artifacts
• Space (conceptual), "location" in a set of categories structured so that "nearby" agents will tend to interact
• Selection, processes that lead to an increase or decrease in the frequency of various types of agent or strategies
• Success criteria or performance measures, a "score" used by an agent or designer in attributing credit in the selection of relatively successful (or unsuccessful) strategies or agents

CAS are occasionally modeled by means of agent-based models and complex network-based models.^[19] Agent-based models are developed by means of various methods and tools primarily by means of first identifying the different agents inside the model.^[20] Another method of developing models for CAS involves developing complex network models by means of using interaction data of various CAS components.^[21]

Recently,^[when?] SpringerOpen/BioMed Central has launched an online open-access journal on the topic of complex adaptive systems modeling (CASM).^[22]

Living organisms are complex adaptive systems. Although complexity is hard to quantify in biology, evolution has produced some remarkably complex organisms.^[23] This observation has led to the common misconception of evolution being progressive and leading towards what are viewed as "higher organisms".^[24]

If this were generally true, evolution would possess an active trend towards complexity. As shown below, in this type of process the value of the most common amount of complexity would increase over time.^[25] Indeed, some artificial life simulations have suggested that the generation of CAS is an inescapable feature of evolution.^[26][27]

However, the idea of a general trend towards complexity in evolution can also be explained through a passive process.^[25] This involves an increase in variance but the most common value, the mode, does not change. Thus, the maximum level of complexity increases over time, but only as an indirect product of there being more organisms in total. This type of random process is also called a bounded random walk.

In this hypothesis, the apparent trend towards more complex organisms is an illusion resulting from concentrating on the small number of large, very complex organisms that inhabit the right-hand tail of the complexity distribution and ignoring simpler and much more common organisms. This passive model emphasizes that the overwhelming majority of species are microscopic prokaryotes,^[28] which comprise about half the world's biomass^[29] and constitute the vast majority of Earth's biodiversity.^[30] Therefore, simple life remains dominant on Earth, and complex life appears more diverse only because of sampling bias.

This lack of an overall trend towards complexity in biology does not preclude the existence of forces driving systems towards complexity in a subset of cases. These minor trends are balanced by other evolutionary pressures that drive systems towards less complex states.

Wikimedia Commons has media related to Complex adaptive systems.

• Complexity Digest comprehensive digest of latest CAS related news and research.
• Complex Adaptive Systems Group loosely coupled group of scientists and software engineers interested in complex adaptive systems
• DNA Wales Research Group Current Research in Organisational change CAS/CES related news and free research data. Also linked to the Business Doctor & BBC documentary series
• A description of complex adaptive systems on the Principia Cybernetica Web.
• Quick reference single-page description of the 'world' of complexity and related ideas hosted by the Center for the Study of Complex Systems at the University of Michigan.
• Complex systems research network
• The Open Agent-Based Modeling Consortium
• A group of multidisciplinary researchers interested in the Modeling and Simulation of Complex Adaptive Systems
• TEDxRotterdam - Igor Nikolic - Complex adaptive systems, and The emergence of universal consciousness: Brendan Hughes at TEDxPretoria . Talks discussing various practical examples of complex adaptive systems, including Wikipedia, star galaxies, genetic mutation, and other examples