Complexity characterises the behaviour of a system or model whose components interact in multiple ways and follow local rules, meaning there is no reasonable higher instruction to define the various possible interactions.^{[1]}
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.
Science as of 2010^{[update]} takes a number of approaches to characterizing complexity; Zayed et al.^{[2]} 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".^{[3]}
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✪ Introduction to Big O Notation and Time Complexity (Data Structures & Algorithms #7)

✪ Living With Complexity

✪ 1.5.1 Time Complexity #1

✪ 23. Computational Complexity

✪ What is Time Complexity Analysis?  Basics of Algorithms
Transcription
Hey everyone. In this video, I'm going to give you an introduction to Big O notation and time complexity. These concepts basically give you one way of describing how the time it takes to run your function grows as the size of the input grows. To see what I mean by that exactly, let's take a look at a few examples here. First of all, let's say you are given an array like this and let's say that this array could be of any lengths. It could be one hundred elements long, a thousand elements long or even one hundred thousand elements. And let's say you want to write a function that takes this array and returns the sum of all the numbers in this array. So, in your function you wanna add up all the numbers of this array and returns the sum. And that function might look like this function right here and I'm gonna use pseudocode here to write this function. So, first of all let's define our function that gonna be called find sum which is going to take given array as input and then inside this function first of all we gonna initialize a variable called total to 0 and then for each i in this given array or for each number total to 0
Contents
Overview
Definitions of complexity often depend on the concept of a confidential "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.^{[4]} 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".^{[4]} Weaver's 1948 paper has influenced subsequent thinking about complexity.^{[5]}
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. organized
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 moreuniform, 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 realworld systems, including planetary orbits, eventually become theoretically unpredictable even using Newtonian dynamics; as discovered by modern chaos theory.^{[6]}
Organized complexity, in Weaver's view, resides in nothing else than the nonrandom, 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 visavis 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.^{[7]}
Sources and factors
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 selforganizing 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.^{[8]}
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 meanings
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, timebounded Kolmogorov complexity, and spacebounded 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.^{[9]} 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 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)^{[10]}^{[11]}. 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^{[12]}, 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 observerindependent measure based only on the internal dynamics of the system, and has been used in studies of emergence and selforganization^{[13]}.
 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,^{[14]} 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.
 In abstract sense – Abstract Complexity, is based on visual structures perception ^{[15]} It is complexity of binary string defined as a square of features number divided by number of elements (0's and 1's). Features comprise here all distinctive arrangements of 0's and 1's. Though the features number have to be always approximated the definition is precise and meet intuitive criterion.
Other fields introduce less precisely defined notions of complexity:
 A complex adaptive system has some or all of the following attributes:^{[3]}
 The number of parts (and types of parts) in the system and the number of relations between the parts is nontrivial – however, there is no general rule to separate "trivial" from "nontrivial";
 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 nontrivial or nonlinear;
 The system can be influenced by, or can adapt itself to, its environment;
 The system is highly sensitive to initial conditions.
Study
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.^{[16]} 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 stock markets, social systems^{[17]}. One such interdisciplinary group of fields is relational order theories.
Topics
Behaviour
The behavior of a complex system is often said to be due to emergence and selforganization. Chaos theory has investigated the sensitivity of systems to variations in initial conditions as one cause of complex behaviour.
Mechanisms
Recent developments around artificial life, evolutionary computation and genetic algorithms have led to an increasing emphasis on complexity and complex adaptive systems.
Simulations
In social science, the study on the emergence of macroproperties from the microproperties, also known as macromicro 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.
Systems
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 sociocognitive systems and emerging systemics research. Complex systems tend to be highdimensional, nonlinear, and difficult to model. In specific circumstances, they may exhibit lowdimensional behaviour.
Data
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 Turingcomplete 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.
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.^{[18]}
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.^{[19]}
Instance hardness is a bottomup 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^{[20]} and could be expanded to other areas.
In molecular recognition
A recent study based on molecular simulations and compliance constants describes molecular recognition as a phenomenon of organisation.^{[21]} 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.
Applications
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.
See also
 Chaos theory
 Command and Control Research Program
 Complex systems
 Complexity theory (disambiguation page)
 Cyclomatic complexity
 Digital morphogenesis
 Dualphase evolution
 Emergence
 Evolution of complexity
 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
 Process architecture
 Programming Complexity
 Sociology and complexity science
 Systems theory
 Thorngate's postulate of commensurate complexity
 Variety (cybernetics)
 Volatility, uncertainty, complexity and ambiguity
 Computational irreducibility
 ZeroForce Evolutionary Law
References
 ^ Johnson, Steven (2001). Emergence: The Connected Lives of Ants, Brains, Cities. New York: Scribner. p. 19. ISBN 9783411040742.
 ^ J. M. Zayed, N. Nouvel, U. Rauwald, O. A. Scherman. Chemical Complexity – supramolecular selfassembly 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
 ^ ^{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 9781780740492. Archived from the original (PDF) on 20151211. Retrieved 20130629.
 ^ ^{a} ^{b} Weaver, Warren (1948). "Science and Complexity" (PDF). American Scientist. 36 (4): 536–44. PMID 18882675. Retrieved 20071121.
 ^ Johnson, Steven (2001). Emergence: the connected lives of ants, brains, cities, and software. New York: Scribner. p. 46. ISBN 9780684868752.
 ^ "Sir James Lighthill and Modern Fluid Mechanics", by Lokenath Debnath, The University of TexasPan American, US, Imperial College Press: ISBN 9781848161139: ISBN 1848161131, Singapore, page 31. Online at http://cs5594.userapi.com/u11728334/docs/25eb2e1350a5/Lokenath_Debnath_Sir_James_Lighthill_and_mode.pdf^{[permanent dead link]}
 ^ Jacobs, Jane (1961). The Death and Life of Great American Cities. New York: Random House.
 ^ Ulanowicz, Robert, "Ecology, the Ascendant Perspective", Columbia, 1997
 ^ 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
 ^ Crutchfield, J.P.; Young, K. (1989). "Inferring statistical complexity". Physical Review Letters. 63: 105–108.
 ^ Crutchfield, J.P.; Shalizi, C.R. (1999). "Thermodynamic depth of causal states: Objective complexity via minimal representations". Physical Review E. 59: 275–283.
 ^ Grassberger, P. (1986). "Toward a quantitative theory of selfgenerated complexity". International Journal of Theoretical Physics. 25: 907. Bibcode:1986IJTP...25..907G. doi:10.1007/bf00668821.
 ^ Prokopenko, M.; Boschetti, F.; Ryan, A. (2009). "An informationtheoretic primer on complexity, selforganisation and emergence". Complexity. 15(1): 11–28.
 ^ 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).
 ^ Mariusz Stanowski (2011) Abstract Complexity Definition, Complicity 2, p.7883 [1]
 ^ Lissack, Michael R.; Johan Roos (2000). The Next Common Sense, The eManager's Guide to Mastering Complexity. Intercultural Press. ISBN 9781857882353.
 ^ BastardasBoada, Albert. "Complexics as a metatransdisciplinary field". Academia.edu.
 ^ 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.
 ^ Smith, M.R.; Martinez, T.; GiraudCarrier, C. (2014). "An Instance Level Analysis of Data Complexity". Machine Learning, 95(2): 225–256.
 ^ Saez, J.; Luengo, J.; Herrera, F. (2013). "Predicting Noise Filtering Efficacy with Data Complexity Measures for Nearest Neighbor Classification". Pattern Recognition 46 (1) pp 355–364.
 ^ Jorg Grunenberg (2011). "Complexity in molecular recognition". Phys. Chem. Chem. Phys. 13 (21): 10136–10146. doi:10.1039/c1cp20097f. PMID 21503359.
Further reading
 Chu, Dominique (2011). "Complexity: Against Systems" (PDF). Theory in Biosciences. 130 (3): 229–45. doi:10.1007/s1206401101214. PMID 21287293.
 Waldrop, M. Mitchell (1992). Complexity: The Emerging Science at the Edge of Order and Chaos. New York: Simon & Schuster. ISBN 9780671767891.
 Czerwinski, Tom; David Alberts (1997). Complexity, Global Politics, and National Security (PDF). National Defense University. ISBN 9781579060466.
 Solé, R. V.; B. C. Goodwin (2002). Signs of Life: How Complexity Pervades Biology. Basic Books. ISBN 9780465019281.
 Heylighen, Francis (2008). "Complexity and SelfOrganization" (PDF). In Bates, Marcia J.; Maack, Mary Niles. Encyclopedia of Library and Information Sciences. CRC. ISBN 9780849397127.
 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 9780387758886
 Mitchell, M. (2009). Complexity: A Guided Tour. Oxford University Press, Oxford, UK.
 Gershenson, C., Ed. (2008). Complexity: 5 Questions. Automatic Peess / VIP.
External links
Wikiquote has quotations related to: Complexity 
Look up complexity in Wiktionary, the free dictionary. 
 Complexity Measures – an article about the abundance of notthatuseful complexity measures.
 Exploring Complexity in Science and Technology – 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