Complex systems: Difference between revisions

{{see also|Complex system|Complex Systems (journal)}}

{{Complex systems}}

'''Complex systems''' present problems both in [[mathematical model]]ling and [[philosophy|philosophical]] foundations. The study of complex systems is an approach to science that investigates how relationships between parts give rise to the collective behaviors of a system and how the system interacts and forms relationships with its environment.<ref>{{cite journal |url=http://www.eolss.net/sample-chapters/c15/E1-29-01-00.pdf |title=General Features of Complex Systems |last=Bar-Yam |first=Yaneer |date=2002 |journal=Encyclopedia of Life Support Systems |publisher=[[Encyclopedia of Life Support Systems|EOLSS]] [[UNESCO]] Publishers, Oxford, UK |accessdate=16 September 2014 |doi= |pmid=}}</ref> The subject is also sometimes called ''complex systems theory'', ''complexity science'', ''study of complex systems'', ''complex networks'', ''network science'', and ''sciences of complexity''.

The equations from which models of complex systems are developed generally derive from [[statistical physics]], [[information theory]], and [[non-linear dynamics]] and represent organized but unpredictable behaviors of natural [[system]]s that are considered fundamentally [[complexity|complex]]. Often times, the physical manifestations of such systems are difficult to define, and so it is common to identify "the system" with the mathematical model rather than referring to the undefined physical subject the model represents. Such a systems approach is often used in [[computer science]], [[biology]],<ref>[[Georges Chapouthier|Chapouthier, G]], Mosaic structures – a working hypothesis for the complexity of living organisms, E-Logos (Electronic Journal for Philosophy), 2009, 17, http://nb.vse.cz/kfil/elogos/biocosmology/chapouthier09.pdf</ref> [[Complexity economics|economics]], [[physics]], [[chemistry]],<ref>J. M. Zayed, N. Nouvel, U. Rauwald, O. A. Scherman, Chemical Complexity – supramolecular self-assembly of synthetic and biological building blocks in water, ''[[Chemical Society Reviews]]'', 2010, 39, 2806–2816 http://pubs.rsc.org/en/Content/ArticleLanding/2010/CS/b922348g</ref> architecture,<ref>C. Alexander, New Concepts in Complexity Theory: : Arising from studies in the field of architecture, an overview of the four books of The Nature of Order with emphasis on the scientific problems which are raised, http://natureoforder.com/library/scientific-introduction.pdf</ref> and many other fields. A variety of abstract [[complex system|theoretical complex systems]] is studied as a field of mathematics.

The key problems of complex systems are difficulties with their formal [[Scientific modelling|modelling]] and [[simulation]]. From such a perspective, in different research contexts complex systems are defined on the basis of their different attributes. Since all complex systems have many interconnected components, the [[Network science|science of networks]] and [[network theory]] are important and useful tools for the study of complex systems. A theory for the resilience of system of systems represented by a network of interdependent networks was developed by Buldyrev et al.<ref name="BuldyrevParshani2010">{{cite journal|last1=Buldyrev|first1=Sergey V.|last2=Parshani|first2=Roni|last3=Paul|first3=Gerald|last4=Stanley|first4=H. Eugene|last5=Havlin|first5=Shlomo|title=Catastrophic cascade of failures in interdependent networks|journal=Nature|volume=464|issue=7291|year=2010|pages=1025–1028|issn=0028-0836|doi=10.1038/nature08932|pmid=20393559|bibcode=2010Natur.464.1025B}}</ref><ref name="ParshaniBuldyrev2010">{{cite journal|last1=Parshani|first1=Roni|last2=Buldyrev|first2=Sergey V.|last3=Havlin|first3=Shlomo|title=Interdependent Networks: Reducing the Coupling Strength Leads to a Change from a First to Second Order Percolation Transition|journal=Physical Review Letters|volume=105|issue=4|pages=048701|year=2010|issn=0031-9007|doi=10.1103/PhysRevLett.105.048701|pmid=20867893|bibcode=2010PhRvL.105d8701P}}</ref> A consensus regarding a single universal definition of ''[[complex system]]'' does not yet exist.

For systems that are less usefully represented with equations various other kinds of narratives and methods for identifying, exploring, designing and interacting with complex systems are used.

== Overview ==

[[File:Braitenberg vehicle (simulation made with breve).jpg|thumb|right|210px|A [[Braitenberg vehicles|Braitenberg]] simulation, programmed in [[breve (software)|breve]], an [[artificial life]] simulator]]

The study of mathematical complex system models is used for many scientific questions poorly suited to the traditional mechanistic conception provided by science.<ref>http://www.narberthpa.com/Bale/lsbale_dop/cybernet.htm Bale, L.S. 1995, ''Gregory Bateson, Cybernetics and the Social/Behavioral Sciences''</ref> ''Complex systems'' is therefore often used as a broad term encompassing a research approach to problems in many diverse disciplines including [[anthropology]], [[artificial intelligence]], [[artificial life]], [[physics]], [[chemistry]], [[computer science]], [[economics]], [[evolutionary computation]], [[earthquake]] prediction, [[meteorology]], [[molecular biology]], [[neuroscience]], [[psychology]] and [[sociology]].

Traditionally, engineering has striven to solve the non-linear system problem while bearing in mind that for small perturbations, most non-linear systems can be approximated with linear systems, significantly simplifying the analysis. Linear systems represent the main class of systems for which general techniques for stability control and analysis exist. However, many physical systems (for example [[lasers]]) are inherently "complex systems" in terms of the definition above, and engineering practice must now include elements of complex systems research.

[[Information theory]] applies well to the [[complex adaptive systems]], CAS, through the concepts of object-oriented design, as well as through formalized concepts of organization and disorder that can be associated with any systems evolution process.

== History ==

[[File:Complexity Map.svg|thumb|360px|A history of complexity science. Update to 2020: http://www.art-sciencefactory.com/complexity-map_feb09.html|alt= http://www.art-sciencefactory.com/complexity-map_feb09.html]]

Complex systems is an approach to science that studies how relationships between parts give rise to the collective behaviors of a [[system]] and how the system interacts and forms relationships with its environment.

The earliest precursor to modern complex systems theory can be found in the classical political economy of the [[Scottish Enlightenment]], later developed by the [[Austrian school of economics]], which says that order in market systems is spontaneous (or [[Emergence|emergent]]) in that it is the result of human action, but not the execution of any human design.<ref>{{cite book |last=Ferguson |first=Adam |authorlink=Adam Ferguson |coauthors= |title=An Essay on the History of Civil Society |publisher=T. Cadell |year=1767 |location=London |pages=Part the Third, Section II, p. 205 |url=http://oll.libertyfund.org/index.php?option=com_staticxt&staticfile=show.php%3Ftitle=1428&Itemid=28 |doi= |id= |isbn= |nopp=true}}</ref><ref>Friedrich Hayek, "The Results of Human Action but Not of Human Design" in ''New Studies in Philosophy, Politics, Economics'', Chicago: University of Chicago Press, 1978, pp. 96–105.</ref>

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 [[Economic calculation problem#Computational complexity|computational complexity]].{{Citation needed|date=November 2016}}

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,<ref>Bruce J. Caldwell, Popper and Hayek: [http://www.unites.uqam.ca/philo/pdf/Caldwell_2003-01.pdf Who influenced whom?], Karl Popper 2002 Centenary Congress, 2002.</ref> not constraining his work to human economies but venturing into other fields such as [[psychology]],<ref>Friedrich von Hayek, ''The Sensory Order: An Inquiry into the Foundations of Theoretical Psychology'', The University of Chicago Press, 1952.</ref> [[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.

The first research institute focused on complex systems, the [[Santa Fe Institute]], was founded in 1984.<ref>{{cite journal | last1 = Ledford | first1 = H | year = 2015 | title = How to solve the world's biggest problems | url = http://www.nature.com/news/how-to-solve-the-world-s-biggest-problems-1.18367 | journal = Nature | volume = 525 | issue = 7569| pages = 308–311 | doi=10.1038/525308a}}</ref> Early Santa Fe Institute participants included physics Nobel laureates [[Murray Gell-Mann]] and [[Philip Warren Anderson|Philip Anderson]], economics Nobel laureate [[Kenneth Arrow]], and Manhattan Project scientists [[George Cowan]] and [[Herbert L. Anderson|Herb Anderson]].<ref>Waldrop, M. M. (1993). [https://books.google.com/books/about/Complexity.html?id=JTRJxYK_tZsC Complexity: The emerging science at the edge of order and chaos.] Simon and Schuster.</ref> Today, there are over 50 [[#Institutes_and_research_centers|institutes and research centers]] focusing on complex systems.

== Typical areas of study ==

===Complexity in practice===

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.

===Complexity management===

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 [[Future Combat Systems|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.<ref>[http://csis.org/files/publication/090410_Organizing_for_a_Complex_World_The_Way_Ahead_0.pdf CSIS paper: "Organizing for a Complex World: The Way Ahead]</ref>

===Complexity economics===

Over the last decades, within the emerging field of [[complexity economics]] new predictive tools have been developed to explain economic growth. Such is the case with the models built by the [[Santa Fe Institute]] in 1989 and the more recent [[economic complexity index]] (ECI), introduced by the [[MIT]] physicist [[Cesar A. Hidalgo]] and the [[Harvard]] economist [[Ricardo Hausmann]]. Based on the ECI, Hausmann, Hidalgo and their team of [[The Observatory of Economic Complexity]] have [[List of countries by future GDP (based on ECI) estimates|produced GDP forecasts for the year 2020]].{{Citation needed|date=February 2016}}

=== Complexity and education ===

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."<ref>{{Cite journal|last=Forsman|first=Jonas|last2=Moll|first2=Rachel|last3=Linder|first3=Cedric|date=2014|title=Extending the theoretical framing for physics education research: An illustrative application of complexity science|url=http://link.aps.org/doi/10.1103/PhysRevSTPER.10.020122|journal=Physical Review Special Topics - Physics Education Research|volume=10|issue=2|doi=10.1103/PhysRevSTPER.10.020122|id=http://hdl.handle.net/10613/2583}}</ref>

===Complexity and modeling===

[[File:Complex-adaptive-system.jpg|300px|right|thumb|A [[complex adaptive system]] model]]

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 [[Scientific modelling|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.<ref>[http://www.reason.com/news/show/33304.html Reason Magazine - The Road from Serfdom<!-- Bot generated title -->]</ref> Hayek would notably explain that complex phenomena, through modeling, can only allow pattern predictions, compared with the precise predictions that can be made out of non-complex phenomena.<ref>[http://nobelprize.org/nobel_prizes/economics/laureates/1974/hayek-lecture.html Friedrich August von Hayek - Prize Lecture<!-- Bot generated title -->]</ref>

[[File:Mathematical models for complex systems.jpg|300px|right|thumb|This is a schematic representation of three types of mathematical models of complex systems with the level of their mechanistic understanding.]]

Mathematical models of complex systems are of three types: [[black-box]] (phenomenological), [[White box (software engineering)|white-box]] (mechanistic, based on the [[first principles]]), and [[Grey box model|grey-box]] (mixtures of phenomenological and mechanistic models) <ref name="Kalmykov Lev V., Kalmykov Vyacheslav L. Solution">

{{Citation

| last = Kalmykov

| first = Lev V.

| last2 = Kalmykov

| first2 = Vyacheslav L.

| title = A Solution to the Biodiversity Paradox by Logical Deterministic Cellular Automata

| journal = Acta Biotheoretica

| pages = 1–19

| year = 2015

| url = http://dx.doi.org/10.1007/s10441-015-9257-9

| doi = 10.1007/s10441-015-9257-9

}}</ref>

.<ref name="Kalmykov Lev V., Kalmykov Vyacheslav L. White-box model">

{{Citation

| last = Kalmykov

| first = Lev V.

| last2 = Kalmykov

| first2 = Vyacheslav L.

| title = A white-box model of S-shaped and double S-shaped single-species population growth

| journal = PeerJ

| volume = 3

| year = 2015

| url = https://dx.doi.org/10.7717/peerj.948

| doi = 10.7717/peerj.948

| pages=e948

| pmid=26038717

| pmc=4451025

}}</ref> In black-box models, the individual-based mechanisms of a complex dynamic system remain hidden. Black-box models are completely nonmechanistic. They are phenomenological and ignore a composition and internal structure of a complex system. We cannot investigate interactions of subsystems of such a non-transparent model. A white-box model of complex dynamic system has ‘transparent walls’ and directly shows underlying mechanisms. All events at micro-, meso- and macro-levels of a dynamic system are directly visible at all stages of its white-box model evolution. In most cases mathematical modelers use the heavy black-box mathematical methods, which cannot produce mechanistic models of complex dynamic systems. Grey-box models are intermediate and combine black-box and white-box approaches. As a rule, this approach is used in ‘overloaded’ form,{{clarify|date=August 2016}} which makes it less transparent. It was demonstrated that the logical deterministic [[cellular automata]] approach allows the creation of white-box models of ecosystems.<ref name="Kalmykov Lev V., Kalmykov Vyacheslav L.">

{{Citation

| last = Kalmykov

| first = Lev V.

| last2 = Kalmykov

| first2 = Vyacheslav L.

| title = Verification and reformulation of the competitive exclusion principle

| journal = Chaos, Solitons & Fractals

| volume = 56

| pages = 124–131

| year = 2013

| url = http://dx.doi.org/10.1016/j.chaos.2013.07.006

| doi = 10.1016/j.chaos.2013.07.006

}}</ref> Creation of a white-box model of complex system is associated with the problem of the necessity of an a priori basic knowledge of the modeling subject. The deterministic logical [[Cellular automaton|cellular automata]] are necessary but not sufficient condition of a white-box model. The second necessary prerequisite of a white-box model is the presence of the physical [[ontology]] of the object under study. The white-box modeling represents an automatic hyper-logical inference from the [[first principle]]s because it is completely based on the deterministic logic and axiomatic theory of the subject. The purpose of the white-box modeling is to derive from the basic axioms a more detailed, more concrete mechanistic knowledge about the dynamics of the object under study. The necessity to formulate an intrinsic [[axiomatic system]] of the subject before creating its white-box model distinguishes the cellular automata models of white-box type from cellular automata models based on arbitrary logical rules. If cellular automata rules have not been formulated from the first principles of the subject, then such a model may have a weak relevance to the real problem.<ref name="Kalmykov Lev V., Kalmykov Vyacheslav L. White-box model" />

===Complexity and chaos theory===

Complexity theory is rooted in [[chaos theory]], which in turn has its origins more than a century ago in the work of the French mathematician [[Henri Poincaré]]. Chaos is sometimes viewed as extremely complicated information, rather than as an absence of order.<ref>Hayles, N. K. (1991). ''Chaos Bound: Orderly Disorder in Contemporary Literature and Science''. Cornell University Press, Ithaca, NY.</ref> Chaotic systems remain deterministic, though their long-term behavior can be difficult to predict with any accuracy. 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<ref>Prigogine, I. (1997). ''The End of Certainty'', The Free Press, New York.</ref> that complexity is non-deterministic, and gives no way whatsoever to precisely predict the future.<ref>See also {{cite journal |author=D. Carfì |year=2008 |title=Superpositions in Prigogine approach to irreversibility |journal=AAPP: Physical, Mathematical, and Natural Sciences |volume=86 |issue=1 |pages=1–13 |url=http://cab.unime.it/journals/index.php/AAPP/article/view/384/0 |format= |accessdate=}}.</ref>

The emergence of complexity theory shows a domain between deterministic order and randomness which is complex.<ref name="PC98">[[Paul Cilliers|Cilliers, P.]] (1998). ''Complexity and Postmodernism: Understanding Complex Systems'', Routledge, London.</ref> This is referred as the "[[edge of chaos]]".<ref>[[Per Bak]] (1996). ''How Nature Works: The Science of Self-Organized Criticality'', Copernicus, New York, U.S.</ref>

[[File:Lorenz attractor yb.svg|thumb|left|200px|A plot of the [[Lorenz attractor]].]]

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,<ref>Colander, D. (2000). ''The Complexity Vision and the Teaching of Economics'', E. Elgar, Northampton, Massachusetts.</ref> 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.<ref name="PC98"/>

Therefore, the main difference between chaotic systems and complex systems is their history.<ref>Buchanan, M. (2000). ''Ubiquity : Why catastrophes happen'', three river press, New-York.</ref> Chaotic systems do not rely on their history as complex ones do. Chaotic behaviour pushes a system in equilibrium into chaotic order, which means, in other words, out of what we traditionally define as 'order'.{{clarify|date=September 2011}} On the other hand, complex systems evolve far from equilibrium at the [[edge of chaos]]. 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."<ref>Gell-Mann, M. (1995). What is Complexity? Complexity 1/1, 16-19</ref> In a sense chaotic systems can be regarded as a subset of complex systems distinguished precisely by this absence of historical dependence. 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.

{{clear left}}

===Complexity and network science===

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.<ref name="DorogovtsevMendes2003">{{cite journal|last1=Dorogovtsev|first1=S.N.|last2=Mendes|first2=J.F.F.|year=2003|doi=10.1093/acprof:oso/9780198515906.001.0001|title=Evolution of Networks}}</ref>

<ref name="Fortunato2011">{{cite journal|last1=Fortunato|first1=Santo|title=Reuven Cohen and Shlomo Havlin: Complex Networks|journal=Journal of Statistical Physics|volume=142|issue=3|year=2011|pages=640–641|issn=0022-4715|doi=10.1007/s10955-011-0129-7}}</ref><ref name="Newman2010">{{cite journal|last1=Newman|first1=Mark|year=2010|doi=10.1093/acprof:oso/9780199206650.001.0001|title=Networks}}</ref> for example, the INTERNET can be represented as a network composed of nodes (computers) and links (direct connections between computers). Other examples are social networks, airline networks <ref name="BarratBarthelemy2004">{{cite journal|last1=Barrat|first1=A.|last2=Barthelemy|first2=M.|last3=Pastor-Satorras|first3=R.|last4=Vespignani|first4=A.|title=The architecture of complex weighted networks|journal=Proceedings of the National Academy of Sciences|volume=101|issue=11|year=2004|pages=3747–3752|issn=0027-8424|doi=10.1073/pnas.0400087101}}</ref>, biological networks and climate networks.<ref name="YamasakiGozolchiani2008">{{cite journal|last1=Yamasaki|first1=K.|last2=Gozolchiani|first2=A.|last3=Havlin|first3=S.|title=Climate Networks around the Globe are Significantly Affected by El Niño|journal=Physical Review Letters|volume=100|issue=22|year=2008|issn=0031-9007|doi=10.1103/PhysRevLett.100.228501}}</ref>

Networks can also fail and recover spontaneously. For modeling this phenomena see ref. <ref name="MajdandzicPodobnik2013">{{cite journal|last1=Majdandzic|first1=Antonio|last2=Podobnik|first2=Boris|last3=Buldyrev|first3=Sergey V.|last4=Kenett|first4=Dror Y.|last5=Havlin|first5=Shlomo|last6=Eugene Stanley|first6=H.|title=Spontaneous recovery in dynamical networks|journal=Nature Physics|volume=10|issue=1|year=2013|pages=34–38|issn=1745-2473|doi=10.1038/nphys2819}}</ref>

Interacting complex systems can be modeled as networks of networks. For their breakdown and recovery properties see <ref name="GaoBuldyrev2011">{{cite journal|last1=Gao|first1=Jianxi|last2=Buldyrev|first2=Sergey V.|last3=Stanley|first3=H. Eugene|last4=Havlin|first4=Shlomo|title=Networks formed from interdependent networks|journal=Nature Physics|volume=8|issue=1|year=2011|pages=40–48|issn=1745-2473|doi=10.1038/nphys2180}}</ref>

<ref name="MajdandzicBraunstein2016">{{cite journal|last1=Majdandzic|first1=Antonio|last2=Braunstein|first2=Lidia A.|last3=Curme|first3=Chester|last4=Vodenska|first4=Irena|last5=Levy-Carciente|first5=Sary|last6=Eugene Stanley|first6=H.|last7=Havlin|first7=Shlomo|title=Multiple tipping points and optimal repairing in interacting networks|journal=Nature Communications|volume=7|year=2016|pages=10850|issn=2041-1723|doi=10.1038/ncomms10850}}</ref>

===General form of complexity computation===

The computational law of reachable optimality<ref>Wenliang Wang (2015). Pooling Game Theory and Public Pension Plan. ISBN 978-1507658246. Chapter 4.</ref> is established as a general form of computation for ordered system and it reveals complexity computation is a compound computation of optimal choice and optimality driven reaching pattern overtime underlying a specific and any experience path of ordered system within the general limitation of system integrity.

The computational law of reachable optimality has four key components as described below.

1. '''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.

2. '''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.

3. '''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.

4. '''Robustness''': The more challenge a reachable optimality can accommodate, the more robust it is in term of path integrity.

There are also four computation features in the law of reachable optimality.

1. '''Optimal Choice''': Computation in realizing Optimal Choice can be very simple or very complex. A simple rule in Optimal Choice is to accept whatever is reached, Reward As You Go (RAYG). A Reachable Optimality computation reduces into optimizing reachability when RAYG is adopted. The Optimal Choice computation can be more complex when multiple NE strategies present in a reached game.

2. '''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.

3. '''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.

4. '''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.

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. When the optimal choice computation involves non-RAYG computation, it's a complexity computation driving the compound effect.

== Notable figures ==

* [[Christopher Alexander]]

* [[Gregory Bateson]]

* [[Ludwig von Bertalanffy]]

* [[Samuel Bowles (economist)|Samuel Bowles]]

* [[Paul Cilliers]]

* [[Murray Gell-Mann]]

* [[Arthur Iberall]]

* [[Stuart Kauffman]]

* [[Cris Moore]]

* [http://www.billmckelvey.org/ Bill McKelvey]

* [[Jerry Sabloff]]

* [[Geoffrey West]]

* [[Yaneer Bar-Yam]]

* [[Walter Clemens, Jr.]]

== See also ==

{{Portal|Systems science}}

{|

|- style="vertical-align:top"

|style="padding-right:2em"|

* [[Chaos theory]]

* [[Cybernetics]]

* [[Cognitive model#Dynamical systems|Cognitive modeling]]

* [[Cognitive Science]]

* [[Complex adaptive system]]

* [[Complex networks]]

* [[Complexity]]

* [[Complexity economics]]

* [[Decision engineering]]

* [[Dual-phase evolution]]

* [[Dynamical system]]

* [[Dynamical systems theory]]

|style="padding-right:2em"|

* [[Emergence]]

* [[Enterprise systems engineering]]

* [[Generative sciences]]

* [[Homeokinetics]]

* [[Interdependent networks]]

* [[Invisible hand]]

* [[Mixed reality]]

* [[Multi-agent system]]

* [[Network Science]]

* [[Nonlinearity]]

* [[Pattern-oriented modeling]]

* [[Percolation Theory]]

* [[Process architecture]]

|

* [[Systems theory]]

** [[Systems theory in anthropology|in anthropology]]

* [[Self-organization]]

* [[Sociology and complexity science]]

* {{longitem|style=line-height:1.35em|[[Volatility, uncertainty, complexity and ambiguity|Volatility, uncertainty, complexity<br/>and ambiguity]]}}

|}

== References ==

{{Reflist}}

== Further reading ==

* Bazin, A. (2014). [https://www.academia.edu/attachments/34737324/download_file?st=MTQxNzA5MzgyNywxMDguMjYuMTIzLjE2MSwxMzMzMjk5MA%3D%3D&s=swp-toolbar&ct=MTQxNzA5MzgyNyw2OTU5MCwxMzMzMjk5MA== Defeating ISIS and Their Complex Way of War] Small Wars Journal.

* Syed M. Mehmud (2011), [http://predictivemodeler.com/sitecontent/book/Ch06_Applications/Actuarial/HEC_Model/Healthcare%20Exchange%20Complexity%20Model%20-%20Report%20-%20Aug2011.pdf ''A Healthcare Exchange Complexity Model'']

* {{cite journal | last1 = Chu | first1 = D. | last2 = Strand | first2 = R. | last3 = Fjelland | first3 = R. | year = 2003 | title = Theories of complexity | url = | journal = Complexity | volume = 8 | issue = 3 | pages = 19–30 | doi = 10.1002/cplx.10059 }}

* [[Luis Amaral|L.A.N. Amaral]] and J.M. Ottino, [http://amaral-lab.org/media/publication_pdfs/Amaral-2004-Eur.Phys.J.B-38-147.pdf ''Complex networks — augmenting the framework for the study of complex system''], 2004.

* {{cite journal | last1 = Gell-Mann | first1 = Murray | year = 1995 | title = Let's Call It Plectics | url = http://www.santafe.edu/~mgm/Site/Publications_files/MGM%20118.pdf | format = PDF | journal = Complexity | volume = 1 | issue = 5 }}

* [[Nigel Goldenfeld]] and Leo P. Kadanoff, [http://guava.physics.uiuc.edu/~nigel/articles/complexity.html ''Simple Lessons from Complexity''], 1999

* A. Gogolin, A. Nersesyan and A. Tsvelik, [http://www.cmth.bnl.gov/~tsvelik/theory.html ''Theory of strongly correlated systems ''], Cambridge University Press, 1999.

* Kelly, K. (1995). [http://www.kk.org/outofcontrol/contents.php ''Out of Control''], Perseus Books Group.

* {{cite journal | last1 = Donald Snooks | first1 = Graeme | year = 2008 | title = A general theory of complex living systems: Exploring the demand side of dynamics | url = | journal = Complexity | volume = 13 | issue = 6 | doi=10.1002/cplx.20225 | pages=12–20}}

* Sorin Solomon and Eran Shir, [http://www.europhysicsnews.org/index.php?option=article&access=standard&Itemid=129&url=/articles/epn/abs/2003/02/epn03204/epn03204.html ''Complexity; a science at 30''], 2003.

* [http://www.oeaw.ac.at/byzanz/repository/Preiser_WorkingPapers_Calculating_I.pdf Preiser-Kapeller, Johannes, "Calculating Byzantium. Social Network Analysis and Complexity Sciences as tools for the exploration of medieval social dynamics". August 2010]

* [[Walter Clemens, Jr.]], [https://web.archive.org/web/20150219221633/http://www.sunypress.edu/p-5782-complexity-science-and-world-af.aspx ''Complexity Science and World Affairs''], SUNY Press, 2013.

== External links ==

{{Commons category|Complex systems}}

{{Wiktionary|complex systems}}

* {{cite web|url=http://www.openabm.org |title=The Open Agent-Based Modeling Consortium}}

* {{cite web|url=http://www.complexity.ecs.soton.ac.uk/ |title=Complexity Science Focus}}

* {{cite web|url=http://www.santafe.edu/ |title=Santa Fe Institute}}

* {{cite web|url=http://indecs.eu/ |title=INDECS}} (Interdisciplinary Description of Complex Systems)

* {{cite web|url=http://www.ccsr.uiuc.edu/ |title=Center for Complex Systems Research, Univ. of Illinois}}

* {{cite web|url=http://havlin.biu.ac.il/course3.php |title=Introduction to complex systems - Short course by Shlomo Havlin}}

* {{cite web|url=http://www.eoearth.org/view/article/51cbed507896bb431f69154d/?topic=51cbfc79f702fc2ba8129ed7 |title=Complex Systems|date=October 24, 2013|author=Jessie Henshaw|publisher=[[Encyclopedia of Earth]]}}

{{Complex systems topics}}

{{Systems science}}

[[Category:Complex systems theory| ]]

[[Category:Cybernetics]]

[[Category:Systems]]

[[Category:Systems science]]

[[Category:Mathematical modeling]]

[[bg:Комплексни системи]]

[[ca:Sistemes complexos]]

[[de:Komplexes System]]

[[es:Sistema complejo]]

[[fr:Système complexe]]

[[it:Sistema complesso]]

[[pt:Sistemas Complexos]]

[[ru:Сложная система]]