In such a system, small changes to initial conditions can lead to dramatically different outcomes. Chaotic behavior can therefore be extremely hard to model numerically, because small rounding errors at an intermediate stage of computation can cause the model to generate completely inaccurate output. Furthermore, if a complex system returns to a state similar to one it held previously, it may behave completely differently in response to exactly the same stimuli, so chaos also poses challenges for extrapolating from past experience.
Another common feature of complex systems is the presence of emergent behaviors and properties: these are traits of a system which are not apparent from its components in isolation but which result from the interactions, dependencies, or relationships they form when placed together in a system. Emergence broadly describes the appearance of such behaviors and properties, and has applications to systems studied in both the social and physical sciences.
While emergence is often used to refer only to the appearance of unplanned organized behavior in a complex system, emergence can also refer to the breakdown of organization; it describes any phenomena which are difficult or even impossible to predict from the smaller entities that make up the system. One example of complex system whose emergent properties have been studied extensively is cellular automata. In a cellular automaton, a grid of cells, each having one of finitely many states, evolves over time according to a simple set of rules.
These rules guide the "interactions" of each cell with its neighbors. Although the rules are only defined locally, they have been shown capable of producing globally interesting behavior, for example in Conway's Game of Life. When emergence describes the appearance of unplanned order, it is spontaneous order in the social sciences or self-organization in physical sciences.
Spontaneous order can be seen in herd behavior , whereby a group of individuals coordinates their actions without centralized planning. Self-organization can be seen in the global symmetry of certain crystals , for instance the apparent radial symmetry of snowflakes , which arises from purely local attractive and repulsive forces both between water molecules and between water molecules and their surrounding environment.
Complex adaptive systems are special cases of complex systems that are 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 , the cities, manufacturing businesses and any human social group-based endeavor in a cultural and social system such as political parties or communities. Complex systems may have the following features: . Although it is arguable that humans have been studying complex systems for thousands of years, the modern scientific study of complex systems is relatively young in comparison to established fields of science 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. The notion of self-organizing systems is tied with work in nonequilibrium thermodynamics , including that pioneered by chemist and Nobel laureate Ilya Prigogine in his study of dissipative structures. Even older is the work by Hartree-Fock on the quantum chemistry equations and later calculations of the structure of molecules which can be regarded as one of the earliest examples of emergence and emergent wholes in science.
One complex system containing humans is the classical political economy of the Scottish Enlightenment , later developed by the Austrian school of economics , which argues that order in market systems is spontaneous or emergent in that it is the result of human action, but not the execution of any human design.
Upon this the Austrian school developed from the 19th to the early 20th century the economic calculation problem , along with the concept of dispersed knowledge , which were to fuel debates against the then-dominant Keynesian economics. This debate would notably lead economists, politicians and other parties to explore the question of computational complexity. A pioneer in the field, and inspired by Karl Popper 's and Warren Weaver 's works, Nobel prize economist and philosopher Friedrich Hayek dedicated much of his work, from early to the late 20th century, to the study of complex phenomena,  not constraining his work to human economies but venturing into other fields such as psychology ,  biology and cybernetics.
Gregory Bateson played a key role in establishing the connection between anthropology and systems theory; he recognized that the interactive parts of cultures function much like ecosystems. While the explicit study of complex systems dates at least to the s,  the first research institute focused on complex systems, the Santa Fe Institute , was founded in The traditional approach to dealing with complexity is to reduce or constrain it. Typically, this involves compartmentalisation: dividing a large system into separate parts.
Organizations, for instance, divide their work into departments that each deal with separate issues. Engineering systems are often designed using modular components. However, modular designs become susceptible to failure when issues arise that bridge the divisions.
As projects and acquisitions become increasingly complex, companies and governments are challenged to find effective ways to manage mega-acquisitions such as the Army Future Combat Systems. Acquisitions such as the FCS rely on a web of interrelated parts which interact unpredictably. As acquisitions become more network-centric and complex, businesses will be forced to find ways to manage complexity while governments will be challenged to provide effective governance to ensure flexibility and resiliency.
Over the last decades, within the emerging field of complexity economics new predictive tools have been developed to explain economic growth. Hidalgo and the Harvard economist Ricardo Hausmann. Focusing on issues of student persistence with their studies, Forsman, Moll and Linder explore the "viability of using complexity science as a frame to extend methodological applications for physics education research", finding that "framing a social network analysis within a complexity science perspective offers a new and powerful applicability across a broad range of PER topics".
One of Friedrich Hayek's main contributions to early complexity theory is his distinction between the human capacity to predict the behaviour of simple systems and its capacity to predict the behaviour of complex systems through modeling. He believed that economics and the sciences of complex phenomena in general, which in his view included biology, psychology, and so on, could not be modeled after the sciences that deal with essentially simple phenomena like physics.
Chaos is sometimes viewed as extremely complicated information, rather than as an absence of order.
With perfect knowledge of the initial conditions and of the relevant equations describing the chaotic system's behavior, one can theoretically make perfectly accurate predictions about the future of the system, though in practice this is impossible to do with arbitrary accuracy. Ilya Prigogine argued  that complexity is non-deterministic, and gives no way whatsoever to precisely predict the future. The emergence of complexity theory shows a domain between deterministic order and randomness which is complex. When one analyzes complex systems, sensitivity to initial conditions, for example, is not an issue as important as it is within chaos theory, in which it prevails.
As stated by Colander,  the study of complexity is the opposite of the study of chaos. Complexity is about how a huge number of extremely complicated and dynamic sets of relationships can generate some simple behavioral patterns, whereas chaotic behavior, in the sense of deterministic chaos, is the result of a relatively small number of non-linear interactions.
Therefore, the main difference between chaotic systems and complex systems is their history. Chaotic behaviour pushes a system in equilibrium into chaotic order, which means, in other words, out of what we traditionally define as 'order'.
They evolve at a critical state built up by a history of irreversible and unexpected events, which physicist Murray Gell-Mann called "an accumulation of frozen accidents". Many real complex systems are, in practice and over long but finite time periods, robust. However, they do possess the potential for radical qualitative change of kind whilst retaining systemic integrity. Metamorphosis serves as perhaps more than a metaphor for such transformations. A complex system is usually composed of many components and their interactions.
Such a system can be represented by a network where nodes represent the components and links represent their interactions. Its resilience to failures was studied using percolation theory. For modeling this phenomenon see Majdandzic et al. For their breakdown and recovery properties see Gao et al. The weighted links represent the velocity between two junctions nodes.
This approach was found useful to characterize the global traffic efficiency in a city. The computational law of reachable optimality  is established as a general form of computation for ordered systems and it reveals complexity computation is a compound computation of optimal choice and optimality driven reaching pattern over time underlying a specific and any experience path of ordered system within the general limitation of system integrity.
Reachability of Optimality : Any intended optimality shall be reachable. Unreachable optimality has no meaning for a member in the ordered system and even for the ordered system itself. Prevailing and Consistency : Maximizing reachability to explore best available optimality is the prevailing computation logic for all members in the ordered system and is accommodated by the ordered system.
Conditionality : Realizable tradeoff between reachability and optimality depends primarily upon the initial bet capacity and how the bet capacity evolves along with the payoff table update path triggered by bet behavior and empowered by the underlying law of reward and punishment. Precisely, it is a sequence of conditional events where the next event happens upon reached status quo from experience path. Robustness : The more challenge a reachable optimality can accommodate, the more robust it is in term of path integrity.
Optimal Choice : Computation in realizing Optimal Choice can be very simple or very complex. The Optimal Choice computation can be more complex when multiple NE strategies present in a reached game. Initial Status : Computation is assumed to start at an interested beginning even the absolute beginning of an ordered system in nature may not and need not present. An assumed neutral Initial Status facilitates an artificial or a simulating computation and is not expected to change the prevalence of any findings. Territory : An ordered system shall have a territory where the universal computation sponsored by the system will produce an optimal solution still within the territory.
Reaching Pattern : The forms of Reaching Pattern in the computation space, or the Optimality Driven Reaching Pattern in the computation space, primarily depend upon the nature and dimensions of measure space underlying a computation space and the law of punishment and reward underlying the realized experience path of reaching. There are five basic forms of experience path we are interested in, persistently positive reinforcement experience path, persistently negative reinforcement experience path, mixed persistent pattern experience path, decaying scale experience path and selection experience path.
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The compound computation in selection experience path includes current and lagging interaction, dynamic topological transformation and implies both invariance and variance characteristics in an ordered system's experience path. In addition, the computation law of reachable optimality gives out the boundary between complexity model, chaotic model and determination model. When RAYG is the Optimal Choice computation, and the reaching pattern is a persistently positive experience path, persistently negative experience path, or mixed persistent pattern experience path, the underlying computation shall be a simple system computation adopting determination rules.
If the reaching pattern has no persistent pattern experienced in RAYG regime, the underlying computation hints there is a chaotic system.
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When the optimal choice computation involves non-RAYG computation, it's a complexity computation driving the compound effect. From Wikipedia, the free encyclopedia. For the journal, see Complex Systems journal. Collective consciousness.
Collective behaviour. Social dynamics Collective intelligence Collective action Self-organized criticality Herd mentality Phase transition Agent-based modelling Synchronization Ant colony optimization Particle swarm optimization Swarm behaviour. Evolution and adaptation.
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Systems theory. Nonlinear dynamics. Game theory. Prisoner's dilemma Rational choice theory Bounded rationality Irrational behaviour Evolutionary game theory. Systems science portal. Volatility, uncertainty, complexity and ambiguity. Encyclopedia of Life Support Systems. Retrieved 16 September Risk and Precaution. Cambridge University Press.
Buldyrev; R. Parshani; G. Paul; H. Stanley; S. Havlin Bibcode : Natur. Scientific Reports. Bibcode : NatSR Models of large systems of simple interacting agents lead to emergent complexity observed in phase transitions - yet, these systems remain at least to a degree tractable by the methods of physics. The wealth and complexity of the associated phenomena have encouraged the drawing of parallels between the physics of phase transitions and the complexity observed in real-life systems across many disciplines of science.
In recent decades, criticality involving a host of associated scaling phenomena has been evoked appropriately or not in the context of a wide range of paradigmatic complex systems, including biological systems such as that of brain dynamics, regulatory systems such as heart rate control, human behaviour, financial and ecological systems and many more. In particular, in recent decades many natural and man-made complex systems have been identified as operating at or near criticality, prompting researchers to suggest analogies between such complex systems and elementary physics phenomena.
Contrary to model physical systems, the exact mechanisms of regulation or self-tuning in these phenomena are often difficult to pinpoint and remain unknown. Regardless of the nature of control and the exact mechanism of tuning towards phase transition, there exists a host of phenomena associated with alteration of a system's dynamics undergoing phase transition. These include an increase in long-range correlations, symmetry breaking, the appearance of soft modes and hard frequencies, and flickering, all constituents of fundamental thermodynamical slowing down.
Paramount, yet under-explored, is the aspect of the predictive role of such phenomena. Indeed, critical slowing down is a profound marker of the approach of a vast change in the system's dynamics. The inhomogeneity, heterogeneity and non-stationarity of the dynamics of critical phase transitions leading to abrupt or gradual critical regime changes in real-life systems are, however, routinely encountered, often yielding these systems intractable. Therefore, even though such phenomena are of profound importance in the characterisation of real-world complexity, still there remains a great divide between 'physics complexity' and real-world complexity, where neither the interactions nor the interacting agents are reducible to simple entities.
Of interest to this research topic are manuscripts dealing with criticality and phase transitions in real-world complex systems. This extends to contributions on criticality of: complex topologies, e. Both spatial and temporal critical phenomena and phase transitions are of interest for this topic. All flavours of work are welcome, from purely theoretical contributions to experimental, simulations and data exploration based work. Review papers, opinion pieces and hypothesis papers are equally welcome.