Complexity characterizes the behavior of a system or model whose components interact in multiple ways and follow local rules, leading to non-linearity, randomness, collective dynamics, hierarchy, and emergence.
[3] As of 2010[update], a number of approaches to characterizing complexity have been used in science; 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".
[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".
[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.
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.
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.
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.
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.
In today's systems, this is the difference between myriad connecting "stovepipes" and effective "integrated" solutions.
Chaos theory has investigated the sensitivity of systems to variations in initial conditions as one cause of complex behaviour.
In social science, the study on the emergence of macro-properties from the micro-properties, also known as macro-micro view in sociology.
These systems are present in the research of a variety of disciplines, including biology, economics, social studies and technology.
These algorithmic measures of complexity tend to assign high values to random noise.
[23] 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.
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.
Eric Chaisson has advanced a cosmological complexity [30] metric which he terms Energy Rate Density.
[31] This approach has been expanded in various works, most recently applied to measuring evolving complexity of nation-states and their growing cities.