Is complexity a word

 ©TECHNOLOGICAL FORECASTING AND SOCIAL CHANGE,  vol. 59, (1998)

What is complexity, asks author-journalist George Johnson in a recent “Science Times,” the science section of The New York Times (May 5, 1997)? Below the headline, “Researchers on Complexity Ponder What It’s All About,” Johnson reports that there is still no agreed-upon definition, much less a theoretically-rigorous formalization, despite the fact that complexity is currently a “hot” research topic. Many books and innumerable scholarly papers have been published on the subject in the past few years, and there is even a new journal, Complexity, devoted to this nascent science. Johnson quotes Dan Stein, chairman of the physics department at the University of Arizona: “Everybody talks about it. [But] in the absence of a good definition, complexity is pretty much in the eye of the beholder.”

This is not to say that the researchers in this area have not been trying to define it. In the 1970s, Gregory Chaitin and Alexei Kolmogorov (independently) pioneered a mathematical measuring-rod that Chaitin called “algorithmic complexity” — that is, the length of the shortest “recipe” for the complete reproduction of a mathematical treatment. The problem with this definition, as Chaitin concedes, is that random sequences are invariably more complex because in each case the recipe is as long as the whole thing being specified; it cannot be “compressed”.

More recently, Charles Bennett has focussed on the concept of “logical depth” — the computational requirements for converting a recipe into a finished product. Though useful, it seems to be limited to processes in which there is a logical structure of some sort. It would seem to exclude the “booming, buzzing confusion” of the real world, where the internal logic may be problematical or only partially knowable — say the immense number of context-specific chaotic interactions that are responsible for producing global weather “patterns”, or the imponderable forces that will determine the future course of the evolutionary process itself.

A number of researchers, especially those who are associated with the Santa Fe Institute, believe that the key lies in the so-called “phase transitions” between highly ordered and highly disordered physical systems. An often-cited analogy is water, whose complex physical properties lie between the highly ordered state of ice crystals and the highly disordered movements of steam molecules. While the “Santa Fe Paradigm” may be useful, it also sets strict limits on what can be termed “complex”. For instance, it seems to exclude the extremes associated with highly ordered or strictly random phenomena, even though there can be more or less complex patterns of order and more or less complex forms of disorder — degrees of complexity that are not associated with phase transitions. (Indeed, random phenomena seem to be excluded by fiat from some definitions of complexity.)

To confuse matters further, a distinction must be made between what could be labelled “objective complexity” — the “embedded” properties of a physical phenomenon and “subjective complexity” — its “meaning” to a human observer. As Timothy Perper has observed (on-line communication), the equation w = f(z) is structurally simple, but it might have a universe of meaning depending upon how its terms are defined. Indeed, information theory is notorious for its reliance on quantitative, statistical measures and its blindness to meaning — where much can be made of very few words. The telephone directory for a large metropolitan area contains many more words than a Shakespeare play, but is it more complex? Furthermore, as Elisabet Sahtouris has pointed out (on-line communication), the degree of complexity that we might impute to a phenomenon can depend upon our frame of reference for viewing it. If we adopt a broad, “ecological” perspective we will see many more factors, and relationships, at work than if we adopt a “physiological” perspective. When Howard Bloom (on-line communication) quotes the line “To see the World in a Grain of Sand…” from William Blake’s famous poem, “Auguries of Innocence”, it reminds us that even a simple object can denote a vast pattern of relationships, if we choose to see it that way. Accordingly, subjective complexity is a highly variable property of the phenomenal world.

Perhaps we need to go back to the semantic drawing-board. Complexity is, after all, a word — a verbal construct, a mental image. Like the words “electron” or “snow” or “blue” or “tree”, complexity is a shorthand tool for thinking and communicating about various aspects of the phenomenal world. Some words may be very narrow in scope. (Presumably all electrons are alike in their basic properties, although their behavior can vary greatly.) However, many other words may hold a potful of meaning. We often use the word “snow” in conversation without taking the trouble to differentiate among the many different kinds of snow, as serious skiers (and Inuit eskimos) routinely do. Similarly, the English word “blue” refers to a broad band of hues in the color spectrum, and we must drape the word with various qualifiers, from navy blue to royal blue to robin’s egg blue (and many more), to denote the subtle differences among them. So it is also, I believe, with the word “complexity”; it is used in many different ways and encompasses a great variety of phenomena. (Indeed, it seems that many theorists, to suit their own purposes, prefer not to define complexity too precisely.)

The “utility” of any word, whether broad or narrow in scope, is always a function of how much information it imparts to the user(s). Take the word “tree”, for example. It tells you about certain fundamental properties that all trees have in common. But it does not tell you whether or not a given tree is deciduous, whether it is tall or short, or even whether it is living or dead. The same shortcoming applies also to the concept of “complexity”. Although there may be some commonalities between a complex personality, a complex wine, a complex piece of music and a complex machine, the similarities are not obvious. Each is complex in a different way, and their complexities cannot be reduced to an all-purpose algorithm. Moreover, the differences among them are at least as important as any common properties.

What in fact does the word “complexity” connote. One of the leaders in the complexity field, Seth Lloyd of MIT, took the trouble to compile a list of some three dozen different ways in which the term is used in scientific discourse. However, this exercise produced no blinding insight. When asked to define complexity, Lloyd told Johnson: “I can’t define it for you, but I know it when I see it.”

Rather than trying to define what complexity is, perhaps it would be more useful to identify the properties that are commonly associated with the term. I would suggest that complexity often (not always) implies the following attributes: (1) a complex phenomenon consists of many parts (or items, or units, or individuals); (2) there are many relationships/interactions among the parts; and (3) the parts produce combined effects (synergies) that are not easily predicted and may often be novel, unexpected, even surprising.

At the risk of inviting the wrath of the researchers in this field, I would argue that complexity per se is one of the less interesting properties of complex phenomena. The differences, and the unique combined properties (synergies) that arise in each case, are vastly more important than the commonalities. If someone does develop a grand, unifying definition-description of complexity, I predict that it will add very little to the tree of knowledge (pardon the pun). But that shouldn’t deter us from trying; the very effort to do so will surely enrich our understanding.

* With thanks to Howard Bloom, Timothy Perper, Elisabet Sahtouris, Peter Frost, Reed Konsler and the pseudonymous Just Mice for a provocative and insightful on-line discussion within Howard Bloom’s “International Paleopsychology Project” group.

Источник

Complexity characterises the behaviour of a system or model whose components interact in multiple ways and follow local rules, leading to nonlinearity, randomness, collective dynamics, hierarchy, and emergence.[1][2]

The term is generally used to characterize something with many parts where those parts interact with each other in multiple ways, culminating in a higher order of emergence greater than the sum of its parts. The study of these complex linkages at various scales is the main goal of complex systems theory.

The intuitive criterion of complexity can be formulated as follows: a system would be more complex if more parts could be distinguished, and if more connections between them existed.[3]

Science as of 2010 takes a number of approaches to characterizing complexity; Zayed et al.[4]
reflect many of these. Neil Johnson states that «even among scientists, there is no unique definition of complexity – and the scientific notion has traditionally been conveyed using particular examples…» Ultimately Johnson adopts the definition of «complexity science» as «the study of the phenomena which emerge from a collection of interacting objects».[5]

OverviewEdit

Definitions of complexity often depend on the concept of a «system» – a set of parts or elements that have relationships among them differentiated from relationships with other elements outside the relational regime. Many definitions tend to postulate or assume that complexity expresses a condition of numerous elements in a system and numerous forms of relationships among the elements. However, what one sees as complex and what one sees as simple is relative and changes with time.

Warren Weaver posited in 1948 two forms of complexity: disorganized complexity, and organized complexity.[6]
Phenomena of ‘disorganized complexity’ are treated using probability theory and statistical mechanics, while ‘organized complexity’ deals with phenomena that escape such approaches and confront «dealing simultaneously with a sizable number of factors which are interrelated into an organic whole».[6] Weaver’s 1948 paper has influenced subsequent thinking about complexity.[7]

The approaches that embody concepts of systems, multiple elements, multiple relational regimes, and state spaces might be summarized as implying that complexity arises from the number of distinguishable relational regimes (and their associated state spaces) in a defined system.

Some definitions relate to the algorithmic basis for the expression of a complex phenomenon or model or mathematical expression, as later set out herein.

Disorganized vs. organizedEdit

One of the problems in addressing complexity issues has been formalizing the intuitive conceptual distinction between the large number of variances in relationships extant in random collections, and the sometimes large, but smaller, number of relationships between elements in systems where constraints (related to correlation of otherwise independent elements) simultaneously reduce the variations from element independence and create distinguishable regimes of more-uniform, or correlated, relationships, or interactions.

Weaver perceived and addressed this problem, in at least a preliminary way, in drawing a distinction between «disorganized complexity» and «organized complexity».

In Weaver’s view, disorganized complexity results from the particular system having a very large number of parts, say millions of parts, or many more. Though the interactions of the parts in a «disorganized complexity» situation can be seen as largely random, the properties of the system as a whole can be understood by using probability and statistical methods.

A prime example of disorganized complexity is a gas in a container, with the gas molecules as the parts. Some would suggest that a system of disorganized complexity may be compared with the (relative) simplicity of planetary orbits – the latter can be predicted by applying Newton’s laws of motion. Of course, most real-world systems, including planetary orbits, eventually become theoretically unpredictable even using Newtonian dynamics; as discovered by modern chaos theory.[8]

Organized complexity, in Weaver’s view, resides in nothing else than the non-random, or correlated, interaction between the parts. These correlated relationships create a differentiated structure that can, as a system, interact with other systems. The coordinated system manifests properties not carried or dictated by individual parts. The organized aspect of this form of complexity vis-a-vis to other systems than the subject system can be said to «emerge,» without any «guiding hand».

The number of parts does not have to be very large for a particular system to have emergent properties. A system of organized complexity may be understood in its properties (behavior among the properties) through modeling and simulation, particularly modeling and simulation with computers. An example of organized complexity is a city neighborhood as a living mechanism, with the neighborhood people among the system’s parts.[9]

Sources and factorsEdit

There are generally rules which can be invoked to explain the origin of complexity in a given system.

The source of disorganized complexity is the large number of parts in the system of interest, and the lack of correlation between elements in the system.

In the case of self-organizing living systems, usefully organized complexity comes from beneficially mutated organisms being selected to survive by their environment for their differential reproductive ability or at least success over inanimate matter or less organized complex organisms. See e.g. Robert Ulanowicz’s treatment of ecosystems.[10]

Complexity of an object or system is a relative property. For instance, for many functions (problems), such a computational complexity as time of computation is smaller when multitape Turing machines are used than when Turing machines with one tape are used. Random Access Machines allow one to even more decrease time complexity (Greenlaw and Hoover 1998: 226), while inductive Turing machines can decrease even the complexity class of a function, language or set (Burgin 2005). This shows that tools of activity can be an important factor of complexity.

Varied meaningsEdit

In several scientific fields, «complexity» has a precise meaning:

  • In computational complexity theory, the amounts of resources required for the execution of algorithms is studied. The most popular types of computational complexity are the time complexity of a problem equal to the number of steps that it takes to solve an instance of the problem as a function of the size of the input (usually measured in bits), using the most efficient algorithm, and the space complexity of a problem equal to the volume of the memory used by the algorithm (e.g., cells of the tape) that it takes to solve an instance of the problem as a function of the size of the input (usually measured in bits), using the most efficient algorithm. This allows classification of computational problems by complexity class (such as P, NP, etc.). An axiomatic approach to computational complexity was developed by Manuel Blum. It allows one to deduce many properties of concrete computational complexity measures, such as time complexity or space complexity, from properties of axiomatically defined measures.
  • In algorithmic information theory, the Kolmogorov complexity (also called descriptive complexity, algorithmic complexity or algorithmic entropy) of a string is the length of the shortest binary program that outputs that string. Minimum message length is a practical application of this approach. Different kinds of Kolmogorov complexity are studied: the uniform complexity, prefix complexity, monotone complexity, time-bounded Kolmogorov complexity, and space-bounded Kolmogorov complexity. An axiomatic approach to Kolmogorov complexity based on Blum axioms (Blum 1967) was introduced by Mark Burgin in the paper presented for publication by Andrey Kolmogorov.[11] The axiomatic approach encompasses other approaches to Kolmogorov complexity. It is possible to treat different kinds of Kolmogorov complexity as particular cases of axiomatically defined generalized Kolmogorov complexity. Instead of proving similar theorems, such as the basic invariance theorem, for each particular measure, it is possible to easily deduce all such results from one corresponding theorem proved in the axiomatic setting. This is a general advantage of the axiomatic approach in mathematics. The axiomatic approach to Kolmogorov complexity was further developed in the book (Burgin 2005) and applied to software metrics (Burgin and Debnath, 2003; Debnath and Burgin, 2003).
  • In information theory, information fluctuation complexity is the fluctuation of information about information entropy. It is derivable from fluctuations in the predominance of order and chaos in a dynamic system and has been used as a measure of complexity in many diverse fields.
  • In information processing, complexity is a measure of the total number of properties transmitted by an object and detected by an observer. Such a collection of properties is often referred to as a state.
  • In physical systems, complexity is a measure of the probability of the state vector of the system. This should not be confused with entropy; it is a distinct mathematical measure, one in which two distinct states are never conflated and considered equal, as is done for the notion of entropy in statistical mechanics.
  • In dynamical systems, statistical complexity measures the size of the minimum program able to statistically reproduce the patterns (configurations) contained in the data set (sequence).[12][13] While the algorithmic complexity implies a deterministic description of an object (it measures the information content of an individual sequence), the statistical complexity, like forecasting complexity,[14] implies a statistical description, and refers to an ensemble of sequences generated by a certain source. Formally, the statistical complexity reconstructs a minimal model comprising the collection of all histories sharing a similar probabilistic future, and measures the entropy of the probability distribution of the states within this model. It is a computable and observer-independent measure based only on the internal dynamics of the system, and has been used in studies of emergence and self-organization.[15]
  • In mathematics, Krohn–Rhodes complexity is an important topic in the study of finite semigroups and automata.
  • In Network theory complexity is the product of richness in the connections between components of a system,[16] and defined by a very unequal distribution of certain measures (some elements being highly connected and some very few, see complex network).
  • In software engineering, programming complexity is a measure of the interactions of the various elements of the software. This differs from the computational complexity described above in that it is a measure of the design of the software.

Other fields introduce less precisely defined notions of complexity:

  • A complex adaptive system has some or all of the following attributes:[5]
    • The number of parts (and types of parts) in the system and the number of relations between the parts is non-trivial – however, there is no general rule to separate «trivial» from «non-trivial»;
    • The system has memory or includes feedback;
    • The system can adapt itself according to its history or feedback;
    • The relations between the system and its environment are non-trivial or non-linear;
    • The system can be influenced by, or can adapt itself to, its environment;
    • The system is highly sensitive to initial conditions.

StudyEdit

Complexity has always been a part of our environment, and therefore many scientific fields have dealt with complex systems and phenomena. From one perspective, that which is somehow complex – displaying variation without being random – is most worthy of interest given the rewards found in the depths of exploration.

The use of the term complex is often confused with the term complicated. In today’s systems, this is the difference between myriad connecting «stovepipes» and effective «integrated» solutions.[17] This means that complex is the opposite of independent, while complicated is the opposite of simple.

While this has led some fields to come up with specific definitions of complexity, there is a more recent movement to regroup observations from different fields to study complexity in itself, whether it appears in anthills, human brains, or economic systems, social systems.[18] One such interdisciplinary group of fields is relational order theories.

TopicsEdit

BehaviourEdit

The behavior of a complex system is often said to be due to emergence and self-organization. Chaos theory has investigated the sensitivity of systems to variations in initial conditions as one cause of complex behaviour.

MechanismsEdit

Recent developments in artificial life, evolutionary computation and genetic algorithms have led to an increasing emphasis on complexity and complex adaptive systems.

SimulationsEdit

In social science, the study on the emergence of macro-properties from the micro-properties, also known as macro-micro view in sociology. The topic is commonly recognized as social complexity that is often related to the use of computer simulation in social science, i.e.: computational sociology.

SystemsEdit

Systems theory has long been concerned with the study of complex systems (in recent times, complexity theory and complex systems have also been used as names of the field). These systems are present in the research of a variety disciplines, including biology, economics, social studies and technology. Recently, complexity has become a natural domain of interest of real world socio-cognitive systems and emerging systemics research. Complex systems tend to be high-dimensional, non-linear, and difficult to model. In specific circumstances, they may exhibit low-dimensional behaviour.

DataEdit

In information theory, algorithmic information theory is concerned with the complexity of strings of data.

Complex strings are harder to compress. While intuition tells us that this may depend on the codec used to compress a string (a codec could be theoretically created in any arbitrary language, including one in which the very small command «X» could cause the computer to output a very complicated string like «18995316»), any two Turing-complete languages can be implemented in each other, meaning that the length of two encodings in different languages will vary by at most the length of the «translation» language – which will end up being negligible for sufficiently large data strings.

These algorithmic measures of complexity tend to assign high values to random noise. However, those studying complex systems would not consider randomness as complexity[who?].

Information entropy is also sometimes used in information theory as indicative of complexity, but entropy is also high for randomness. Information fluctuation complexity, fluctuations of information about entropy, does not consider randomness to be complex and has been useful in many applications.

Recent work in machine learning has examined the complexity of the data as it affects the performance of supervised classification algorithms. Ho and Basu present a set of complexity measures for binary classification problems.[19]

The complexity measures broadly cover:

  • the overlaps in feature values from differing classes.
  • the separability of the classes.
  • measures of geometry, topology, and density of manifolds. Instance hardness is another approach seeks to characterize the data complexity with the goal of determining how hard a data set is to classify correctly and is not limited to binary problems.[20]

Instance hardness is a bottom-up approach that first seeks to identify instances that are likely to be misclassified (or, in other words, which instances are the most complex). The characteristics of the instances that are likely to be misclassified are then measured based on the output from a set of hardness measures. The hardness measures are based on several supervised learning techniques such as measuring the number of disagreeing neighbors or the likelihood of the assigned class label given the input features. The information provided by the complexity measures has been examined for use in meta-learning to determine for which data sets filtering (or removing suspected noisy instances from the training set) is the most beneficial[21] and could be expanded to other areas.

In molecular recognitionEdit

A recent study based on molecular simulations and compliance constants describes molecular recognition as a phenomenon of organisation.[22]
Even for small molecules like carbohydrates, the recognition process can not be predicted or designed even assuming that each individual hydrogen bond’s strength is exactly known.

The law of requisite complexityEdit

Driving from the law of requisite variety, Boisot and McKelvey formulated the ‘Law of Requisite Complexity’, that holds that, in order to be efficaciously adaptive, the internal complexity of a system must match the external complexity it confronts.[23]

Positive, appropriate and negative complexityEdit

The application in project management of the Law of Requisite Complexity, as proposed by Stefan Morcov, is the analysis of positive, appropriate and negative complexity.[24][25]

In project managementEdit

Project complexity is the property of a project which makes it difficult to understand, foresee, and keep under control its overall behavior, even when given reasonably complete information about the project system.[26][27]

In systems engineeringEdit

Maik Maurer considers complexity as a reality in engineering. He proposed a methodology for managing complexity in systems engineering [28]:

                             1.           Define the system.

                             2.           Identify the type of complexity.

                             3.           Determine the strategy.

                             4.           Determine the method.

                             5.           Model the system.

                             6.           Implement the method.

ApplicationsEdit

Computational complexity theory is the study of the complexity of problems – that is, the difficulty of solving them. Problems can be classified by complexity class according to the time it takes for an algorithm – usually a computer program – to solve them as a function of the problem size. Some problems are difficult to solve, while others are easy. For example, some difficult problems need algorithms that take an exponential amount of time in terms of the size of the problem to solve. Take the travelling salesman problem, for example. It can be solved in time   (where n is the size of the network to visit – the number of cities the travelling salesman must visit exactly once). As the size of the network of cities grows, the time needed to find the route grows (more than) exponentially.

Even though a problem may be computationally solvable in principle, in actual practice it may not be that simple. These problems might require large amounts of time or an inordinate amount of space. Computational complexity may be approached from many different aspects. Computational complexity can be investigated on the basis of time, memory or other resources used to solve the problem. Time and space are two of the most important and popular considerations when problems of complexity are analyzed.

There exist a certain class of problems that although they are solvable in principle they require so much time or space that it is not practical to attempt to solve them. These problems are called intractable.

There is another form of complexity called hierarchical complexity. It is orthogonal to the forms of complexity discussed so far, which are called horizontal complexity.

Emerging Applications in Other FieldsEdit

The concept of complexity is being increasingly used in the study of Cosmology, Big History, and Cultural Evolution with increasing granularity, as well as increasing quantification.

Application in CosmologyEdit

Eric Chaisson advanced a cosmoglogical complexity [29] metric he terms Energy Rate Density.[30] This approach has been expanded in various works, most recently applied to measuring evolving complexity of nation-states and their growing cities.[31]

See alsoEdit

  • Assembly theory
  • Chaos theory
  • Complexity theory (disambiguation page)
  • Complex network
  • Complex system
  • Cyclomatic complexity
  • Digital morphogenesis
  • Dual-phase evolution
  • Emergence
  • Evolution of complexity
  • Fractal
  • Game complexity
  • Holism in science
  • Law of Complexity/Consciousness
  • Model of hierarchical complexity
  • Names of large numbers
  • Network science
  • Network theory
  • Novelty theory
  • Occam’s razor
  • Percolation theory
  • Process architecture
  • Programming Complexity
  • Sociology and complexity science
  • Systems theory
  • Thorngate’s postulate of commensurate complexity
  • Variety (cybernetics)
  • Volatility, uncertainty, complexity and ambiguity
  • Arthur Winfree
  • Computational irreducibility
  • Zero-Force Evolutionary Law
  • Project complexity

ReferencesEdit

  1. ^ Johnson, Steven (2001). Emergence: The Connected Lives of Ants, Brains, Cities. New York: Scribner. p. 19. ISBN 978-3411040742.
  2. ^ «What is complex systems science? | Santa Fe Institute». www.santafe.edu. Archived from the original on 2022-04-14. Retrieved 2022-04-17.
  3. ^ Heylighen, Francis (1999). The Growth of Structural and Functional Complexity during Evolution, in; F. Heylighen, J. Bollen & A. Riegler (Eds.) The Evolution of Complexity. (Kluwer Academic, Dordrecht): 17–44.
  4. ^
    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
  5. ^ a b Johnson, Neil F. (2009). «Chapter 1: Two’s company, three is complexity» (PDF). Simply complexity: A clear guide to complexity theory. Oneworld Publications. p. 3. ISBN 978-1780740492. Archived from the original (PDF) on 2015-12-11. Retrieved 2013-06-29.
  6. ^ a b Weaver, Warren (1948). «Science and Complexity» (PDF). American Scientist. 36 (4): 536–44. PMID 18882675. Archived from the original (PDF) on 2009-10-09. Retrieved 2007-11-21.
  7. ^ Johnson, Steven (2001). Emergence: the connected lives of ants, brains, cities, and software. New York: Scribner. p. 46. ISBN 978-0-684-86875-2.
  8. ^ «Sir James Lighthill and Modern Fluid Mechanics», by Lokenath Debnath, The University of Texas-Pan American, US, Imperial College Press: ISBN 978-1-84816-113-9: ISBN 1-84816-113-1, Singapore, page 31. Online at http://cs5594.userapi.com/u11728334/docs/25eb2e1350a5/Lokenath_Debnath_Sir_James_Lighthill_and_mode.pdf[permanent dead link]
  9. ^ Jacobs, Jane (1961). The Death and Life of Great American Cities. New York: Random House.
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  16. ^ A complex network analysis example: «Complex Structures and International Organizations» (Grandjean, Martin (2017). «Analisi e visualizzazioni delle reti in storia. L’esempio della cooperazione intellettuale della Società delle Nazioni». Memoria e Ricerca (2): 371–393. doi:10.14647/87204. See also: French version).
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  19. ^ Ho, T.K.; Basu, M. (2002). «Complexity Measures of Supervised Classification Problems». IEEE Transactions on Pattern Analysis and Machine Intelligence 24 (3), pp 289–300.
  20. ^ Smith, M.R.; Martinez, T.; Giraud-Carrier, C. (2014). «An Instance Level Analysis of Data Complexity». Machine Learning, 95(2): 225–256.
  21. ^ Sáez, José A.; Luengo, Julián; Herrera, Francisco (2013). «Predicting Noise Filtering Efficacy with Data Complexity Measures for Nearest Neighbor Classification». Pattern Recognition. 46 (1): 355–364. Bibcode:2013PatRe..46..355S. doi:10.1016/j.patcog.2012.07.009.
  22. ^ Jorg Grunenberg (2011). «Complexity in molecular recognition». Phys. Chem. Chem. Phys. 13 (21): 10136–10146. Bibcode:2011PCCP…1310136G. doi:10.1039/c1cp20097f. PMID 21503359.
  23. ^ Boisot, M.; McKelvey, B. (2011). «Complexity and organization-environment relations: revisiting Ashby’s law of requisite variety». P. Allen, the Sage Handbook of Complexity and Management: 279–298.
  24. ^ Morcov, Stefan; Pintelon, Liliane; Kusters, Rob J. (2020). «IT Project Complexity Management Based on Sources and Effects: Positive, Appropriate and Negative» (PDF). Proceedings of the Romanian Academy — Series A. 21 (4): 329–336. Archived (PDF) from the original on 2020-12-30.
  25. ^ Morcov, S. (2021). Managing Positive and Negative Complexity: Design and Validation of an IT Project Complexity Management Framework. KU Leuven University. Available at https://lirias.kuleuven.be/retrieve/637007 Archived 2021-11-07 at the Wayback Machine
  26. ^ Marle, Franck; Vidal, Ludovic‐Alexandre (2016). Managing Complex, High Risk Projects — A Guide to Basic and Advanced Project Management. London: Springer-Verlag.
  27. ^ Morcov, Stefan; Pintelon, Liliane; Kusters, Rob J. (2020). «Definitions, characteristics and measures of IT Project Complexity — a Systematic Literature Review» (PDF). International Journal of Information Systems and Project Management. 8 (2): 5–21. doi:10.12821/ijispm080201. S2CID 220545211. Archived (PDF) from the original on 2020-07-11.
  28. ^ Maurer, Maik (2017). Complexity management in engineering design — a primer. Berlin, Germany. ISBN 978-3-662-53448-9. OCLC 973540283.
  29. ^ Chaisson Eric J. 2002. Cosmic Evolution — the Rise of Complexity in Nature. Harvard University Press.https://www.worldcat.org/title/1023218202
  30. ^ Chaisson, Eric J.. “Energy rate density. II. Probing further a new complexity metric.” Complex. 17 (2011): 44-63.https://onlinelibrary.wiley.com/doi/10.1002/cplx.20373 , https://lweb.cfa.harvard.edu/~ejchaisson/reprints/EnergyRateDensity_II_galley_2011.pdf
  31. ^ Chaisson, Eric J. «Energy Budgets of Evolving Nations and Their Growing Cities.» Energies 15, no. 21 (2022): 8212.https://doi.org/10.3390/en15218212 https://www.mdpi.com/1996-1073/15/21/8212/pdf

Further readingEdit

  • Chu, Dominique (2011). «Complexity: Against Systems» (PDF). Theory in Biosciences. 130 (3): 229–45. doi:10.1007/s12064-011-0121-4. PMID 21287293. S2CID 14903039.
  • Waldrop, M. Mitchell (1992). Complexity: The Emerging Science at the Edge of Order and Chaos. New York: Simon & Schuster. ISBN 978-0-671-76789-1.
  • Czerwinski, Tom; David Alberts (1997). Complexity, Global Politics, and National Security (PDF). National Defense University. ISBN 978-1-57906-046-6.
  • Solé, R. V.; B. C. Goodwin (2002). Signs of Life: How Complexity Pervades Biology. Basic Books. ISBN 978-0-465-01928-1.
  • Heylighen, Francis (2008). «Complexity and Self-Organization» (PDF). In Bates, Marcia J.; Maack, Mary Niles (eds.). Encyclopedia of Library and Information Sciences. CRC. ISBN 978-0-8493-9712-7. Archived from the original (PDF) on 2008-03-08. Retrieved 2007-10-19.
  • Burgin, M. (1982) Generalized Kolmogorov complexity and duality in theory of computations, Notices of the Russian Academy of Sciences, v.25, No. 3, pp. 19–23
  • Meyers, R.A., (2009) «Encyclopedia of Complexity and Systems Science», ISBN 978-0-387-75888-6
  • Mitchell, M. (2009). Complexity: A Guided Tour. Oxford University Press, Oxford, UK.
  • Gershenson, C., Ed. (2008). Complexity: 5 Questions. Automatic Peess / VIP.

External linksEdit

Look up complexity in Wiktionary, the free dictionary.

  • Complexity Measures – an article about the abundance of not-that-useful complexity measures.
  • Exploring Complexity in Science and Technology Archived 2011-03-05 at the Wayback Machine – Introductory complex system course by Melanie Mitchell
  • Santa Fe Institute focusing on the study of complexity science: Lecture Videos
  • UC Four Campus Complexity Videoconferences – Human Sciences and Complexity
Источник

1

: something complex

the complexities of the murder trial

2

: the quality or state of being complex

the complexity of the contract

Synonyms

Example Sentences



He was impressed by the complexity of the music.



The diagram illustrates the complexity of the cell’s structure.



He doesn’t grasp the complexity of the situation.

Recent Examples on the Web

For critics who might have doubted whether a teenager could take on a magnum opus of such emotional complexity, this was their answer.


—Leena Kim, Town & Country, 30 Mar. 2023




Remarkable for its combination of complexity and freshness, Bollinger R.D. 2008 offers fine perlage and aromas of peach, yuzu, just-baked brioche, and slivered almonds.


—Mike Desimone And Jeff Jenssen, Robb Report, 28 Mar. 2023




Influencers are often paid as a percentage of their total sales, and product returns add a lot of complexity to this process.


—WIRED, 28 Mar. 2023




The conversation revealed that many of the concerns about AI are rooted in the fear of humans losing their creative ownership with the complexity and automation of AI systems.


—Zenger News, Forbes, 27 Mar. 2023




And one way to deal with that complexity is just to willfully ignore parts of it.


—Steven Strogatz, Quanta Magazine, 22 Mar. 2023




There’s a lot of technological complexity that happens in the background.


—Fortune Editors, Fortune, 22 Mar. 2023




In a sign of the complexity of the case, the New York Fire Department reported Wednesday that a fire erupted in Guo’s penthouse at the Sherry-Netherland around noon, hours after he was arrested.


—Evan Osnos, The New Yorker, 16 Mar. 2023




However, where the roundworm has only 302 neurons, a fruit fly larva has nearly 10 times that amount and represents a higher level of neurological complexity overall.


—Darren Orf, Popular Mechanics, 10 Mar. 2023


See More

These examples are programmatically compiled from various online sources to illustrate current usage of the word ‘complexity.’ Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Send us feedback about these examples.

Word History

Etymology

complex entry 2 + -ity, perhaps after French complexité

First Known Use

1661, in the meaning defined at sense 1

Time Traveler

The first known use of complexity was
in 1661

Dictionary Entries Near complexity

Cite this Entry

“Complexity.” Merriam-Webster.com Dictionary, Merriam-Webster, https://www.merriam-webster.com/dictionary/complexity. Accessed 14 Apr. 2023.

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More from Merriam-Webster on complexity

Last Updated:
7 Apr 2023
— Updated example sentences

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Merriam-Webster unabridged

Источник

сложность, запутанность, запутанное дело

существительное ↓

- сложность; запутанность
- что-л. сложное

a motor-car was a complexity far beyond her mechanical skill — умение чинить такую сложную технику, как автомобиль, было выше её возможностей 

Мои примеры

Словосочетания

the monumental complexity of the issue — монументальная сложность данного вопроса  
the awesome complexity of the universe — потрясающая сложность Вселенной  
rules of byzantine complexity — архисложные, причудливо-запутанные правила  
processing complexity — технологическая сложность  
degree of complexity — степень сложности  
level of complexity — уровень сложности  
complexity class — класс сложности  
operation complexity — тех. сложность эксплуатации  
computational complexity — сложность вычисления  
connection complexity — сложность связи (программных модулей)  
cyclomatic complexity — т.граф. цикломатическая сложность  
distributed complexity — распределенная сложность  

Примеры с переводом

He doesn’t grasp the complexity of the situation. 

Он не понял всей сложности ситуации.

He was impressed by the complexity of the music. 

Он был поражён сложностью этой музыки.

He enjoyed the complexity of modern computers. 

Ему очень нравилась сложность современных компьютеров.

The second major item was of equally beastly complexity. 

Второй важный вопрос был так же ужасно сложен.

I don’t think you appreciate the complexity of the situation. 

Вряд ли вы осознаёте всю сложность данной ситуации.

The complexities of economics are clearly explained. 

Тонкости экономики чётко объяснены.

The diagram illustrates the complexity of the cell’s structure. 

Эта схема иллюстрирует сложность клеточной структуры.

I’m not sure you have an appreciation of the complexity of the situation. 

Я не уверен, что у вас есть понимание всей сложности ситуации.

In a broadband, work may vary in complexity and require a wide range of skills. 

Отдельная работа, выполняемая в рамках широкого тарифного разряда может иметь различную сложность и требовать разного уровня профессиональных навыков.

Примеры, ожидающие перевода

There is increasing recognition of the complexity of the causes of poverty.  

Для того чтобы добавить вариант перевода, кликните по иконке ☰, напротив примера.

Формы слова

noun
ед. ч.(singular): complexity
мн. ч.(plural): complexities

Источник

Princeton’s WordNetRate this definition:2.0 / 1 vote

  1. complexity, complexnessnoun

    the quality of being intricate and compounded

    «he enjoyed the complexity of modern computers»

WiktionaryRate this definition:0.0 / 0 votes

  1. complexitynoun

    The state of being complex; intricacy; entanglement.

  2. complexitynoun

    That which is and renders complex; intricacy; complication.

WikipediaRate this definition:0.0 / 0 votes

  1. Complexity

    Complexity characterises the behaviour of a system or model whose components interact in multiple ways and follow local rules, leading to nonlinearity, randomness, collective dynamics, hierarchy, and emergence.The term is generally used to characterize something with many parts where those parts interact with each other in multiple ways, culminating in a higher order of emergence greater than the sum of its parts. The study of these complex linkages at various scales is the main goal of complex systems theory.
    The intuitive criterion of complexity can be formulated as follows: a system would be more complex if more parts could be distinguished, and if more connections between them existed.Science as of 2010 takes a number of approaches to characterizing complexity; Zayed et al.
    reflect many of these. Neil Johnson states that «even among scientists, there is no unique definition of complexity – and the scientific notion has traditionally been conveyed using particular examples…» Ultimately Johnson adopts the definition of «complexity science» as «the study of the phenomena which emerge from a collection of interacting objects».

Webster DictionaryRate this definition:5.0 / 1 vote

  1. Complexitynoun

    the state of being complex; intricacy; entanglement

  2. Complexitynoun

    that which is complex; intricacy; complication

  3. Etymology: [Cf. F. complexit.]

FreebaseRate this definition:0.0 / 0 votes

  1. Complexity

    In general usage, complexity tends to be used to characterize something with many parts in intricate arrangement. The study of these complex linkages is the main goal of complex systems theory.
    In science there are at this time a number of approaches to characterizing complexity, many of which are reflected in this article. Neil Johnson describes complexity science as the study of the phenomena which emerge from a collection of interacting objects.
    In a business context, complexity management is the methodology to minimize value-destroying complexity and efficiently control value-adding complexity in a cross-functional approach.

British National Corpus

  1. Nouns Frequency

    Rank popularity for the word ‘complexity’ in Nouns Frequency: #1698

How to pronounce complexity?

How to say complexity in sign language?

Numerology

  1. Chaldean Numerology

    The numerical value of complexity in Chaldean Numerology is: 5

  2. Pythagorean Numerology

    The numerical value of complexity in Pythagorean Numerology is: 7

Examples of complexity in a Sentence

  1. Danny Milisavljevic:

    We’ve nicknamed it the Green Monster in honor of Fenway Park in Boston. If you look closely, you’ll notice that it’s pockmarked with what look like mini-bubbles, the shape and complexity are unexpected and challenging to understand.

  2. Clark Bunting:

    Our national parks are a reflection of our nation, both past and present, and these seven new national park sites will further tell America’s stories, bipartisan, congressional approval for protecting and preserving Harriet Tubman’s heroic life and work, Columbian mammoths and Ice Age fossils at Tule Springs, and the complexity of the Manhattan Project continue to make our National Park System our country’s best idea.

  3. Chris Dougherty:

    War is hard. It’s really complex, everything is interacting with everything else, in really hard-to-predict ways. And the important thing is to use that complexity to Chris Dougherty advantage, to make it hard for Chris Dougherty opponent.

  4. Jonty Rushforth:

    Given the increasing complexity of the physical market and changing trade flows, it is ever more important for participants to have clarity on the value of LNG.

  5. Swiss Attorney General Michael Lauber:

    Our investigation is of great complexity and quite substantial. To give you an example, the OAG (Office of the Attorney General) has seized around nine terabytes of data, so far our investigative team obtained evidence concerning 104 banking relations. And be aware that every banking relation represents several bank accounts.

Popularity rank by frequency of use

Translations for complexity

From our Multilingual Translation Dictionary

  • تعقيدArabic
  • обърканост, сложностBulgarian
  • complexitatCatalan, Valencian
  • složitostCzech
  • indviklethed, kompleksitetDanish
  • Komplexität, Schwierigkeit, VielfaltGerman
  • περιπλοκότητα, περιπλοκήGreek
  • complejidadSpanish
  • keerukus, keerulisusEstonian
  • mutkikkuus, vaikeus, hankaluus, ongelma, monimutkaisuusFinnish
  • complexitéFrench
  • castachtIrish
  • complexidadeGalician
  • מורכבותHebrew
  • जटिलताHindi
  • komplikáció, bonyolultság, bonyodalomHungarian
  • բարդությունArmenian
  • complessitàItalian
  • 複雑さJapanese
  • painumas, sudėtingumasLithuanian
  • sarežģītība, komplicētībaLatvian
  • verwikkeling, gecompliceerdheid, complexiteit, ingewikkeldheidDutch
  • forvikling, flokeNorwegian
  • zawiłość, złożonośćPolish
  • complexidadePortuguese
  • complexitateRomanian
  • сложностьRussian
  • zložitosťSlovak
  • zapletenostSlovene
  • krånglighet, komplexitet, komplikationSwedish
  • karmaşıklık, karmaşıkTurkish
  • складністьUkrainian

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