A complex system is a system composed of many components which may interact with each other.[1] Examples of complex systems are Earth's global climate, organisms, the human brain, infrastructure such as power grid, transportation or communication systems, complex software and electronic systems, social and economic organizations (like cities), an ecosystem, a living cell, and, ultimately, for some authors, the entire universe.[2][3][4]
Complex systems are systems whose behavior is intrinsically difficult to model due to the dependencies, competitions, relationships, or other types of interactions between their parts or between a given system and its environment.[5] Systems that are "complex" have distinct properties that arise from these relationships, such as nonlinearity, emergence, spontaneous order, adaptation, and feedback loops, among others.[6] Because such systems appear in a wide variety of fields, the commonalities among them have become the topic of their independent area of research. In many cases, it is useful to represent such a system as a network where the nodes represent the components and links to their interactions.
The term complex systems often refers to the study of complex systems, which is an approach to science that investigates how relationships between a system's parts give rise to its collective behaviors and how the system interacts and forms relationships with its environment.[7] The study of complex systems regards collective, or system-wide, behaviors as the fundamental object of study; for this reason, complex systems can be understood as an alternative paradigm to reductionism, which attempts to explain systems in terms of their constituent parts and the individual interactions between them.
Graphical representation of alternative stable states and the direction of critical slowing down prior to a critical transition (taken from Lever et al. 2020).[13] Top panels (a) indicate stability landscapes at different conditions. Middle panels (b) indicate the rates of change akin to the slope of the stability landscapes, and bottom panels (c) indicate a recovery from a perturbation towards the system's future state (c.I) and in another direction (c.II).
Critical transitions are abrupt shifts in the state of ecosystems, the climate, financial and economic systems or other complex systems that may occur when changing conditions pass a critical or bifurcation point.[14][15][16][17] The 'direction of critical slowing down' in a system's state space may be indicative of a system's future state after such transitions when delayed negative feedbacks leading to oscillatory or other complex dynamics are weak.[13]
The components of a complex system may themselves be complex systems. For example, an economy is made up of organisations, which are made up of people, which are made up of cells – all of which are complex systems. The arrangement of interactions within complex bipartite networks may be nested as well. More specifically, bipartite ecological and organisational networks of mutually beneficial interactions were found to have a nested structure.[18][19] This structure promotes indirect facilitation and a system's capacity to persist under increasingly harsh circumstances as well as the potential for large-scale systemic regime shifts.[20][21]
Dynamic network of multiplicity
As well as coupling rules, the dynamic network of a complex system is important. Small-world or scale-free networks[22][23] which have many local interactions and a smaller number of inter-area connections are often employed. Natural complex systems often exhibit such topologies. In the human cortex for example, we see dense local connectivity and a few very long axon projections between regions inside the cortex and to other brain regions.
May produce emergent phenomena
Complex systems may exhibit behaviors that are emergent, which is to say that while the results may be sufficiently determined by the activity of the systems' basic constituents, they may have properties that can only be studied at a higher level. For example, empirical food webs display regular, scale-invariant features across aquatic and terrestrial ecosystems when studied at the level of clustered 'trophic' species.[24][25] Another example is offered by the termites in a mound which have physiology, biochemistry and biological development at one level of analysis, whereas their social behavior and mound building is a property that emerges from the collection of termites and needs to be analyzed at a different level.
Relationships are non-linear
In practical terms, this means a small perturbation may cause a large effect (see butterfly effect), a proportional effect, or even no effect at all. In linear systems, the effect is always directly proportional to cause. See nonlinearity.
Relationships contain feedback loops
Both negative (damping) and positive (amplifying) feedback are always found in complex systems. The effects of an element's behavior are fed back in such a way that the element itself is altered.
History
In 1948, Dr. Warren Weaver published an essay on "Science and Complexity",[26] exploring the diversity of problem types by contrasting problems of simplicity, disorganized complexity, and organized complexity. Weaver described these as "problems which involve dealing simultaneously with a sizable number of factors which are interrelated into an organic whole."
While the explicit study of complex systems dates at least to the 1970s,[27] the first research institute focused on complex systems, the Santa Fe Institute, was founded in 1984.[28][29] Early Santa Fe Institute participants included physics Nobel laureates Murray Gell-Mann and Philip Anderson, economics Nobel laureate Kenneth Arrow, and Manhattan Project scientists George Cowan and Herb Anderson.[30] Today, there are over 50 institutes and research centers focusing on complex systems.[citation needed]
Since the late 1990s, the interest of mathematical physicists in researching economic phenomena has been on the rise. The proliferation of cross-disciplinary research with the application of solutions originated from the physics epistemology has entailed a gradual paradigm shift in the theoretical articulations and methodological approaches in economics, primarily in financial economics. The development has resulted in the emergence of a new branch of discipline, namely "econophysics", which is broadly defined as a cross-discipline that applies statistical physics methodologies which are mostly based on the complex systems theory and the chaos theory for economics analysis.[31]
The traditional approach to dealing with complexity is to reduce or constrain it. Typically, this involves compartmentalization: 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 of cities
Jane Jacobs described cities as being a problem in organized complexity in 1961, citing Dr. Weaver's 1948 essay.[33] As an example, she explains how an abundance of factors interplay into how various urban spaces lead to a diversity of interactions, and how changing those factors can change how the space is used, and how well the space supports the functions of the city. She further illustrates how cities have been severely damaged when approached as a problem in simplicity by replacing organized complexity with simple and predictable spaces, such as Le Corbusier's "Radiant City" and Ebenezer Howard's "Garden City". Since then, others have written at length on the complexity of cities.[34]
Recurrence quantification analysis has been employed to detect the characteristic of business cycles and economic development. To this end, Orlando et al.[35] developed the so-called recurrence quantification correlation index (RQCI) to test correlations of RQA on a sample signal and then investigated the application to business time series. The said index has been proven to detect hidden changes in time series. Further, Orlando et al.,[36] over an extensive dataset, shown that recurrence quantification analysis may help in anticipating transitions from laminar (i.e. regular) to turbulent (i.e. chaotic) phases such as USA GDP in 1949, 1953, etc. Last but not least, it has been demonstrated that recurrence quantification analysis can detect differences between macroeconomic variables and highlight hidden features of economic dynamics.
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".[37]
Complexity in healthcare research and practice
Healthcare systems are prime examples of complex systems, characterized by interactions among diverse stakeholders, such as patients, providers, policymakers, and researchers, across various sectors like health, government, community, and education. These systems demonstrate properties like non-linearity, emergence, adaptation, and feedback loops.[38] Complexity science in healthcare frames knowledge translation as a dynamic and interconnected network of processes—problem identification, knowledge creation, synthesis, implementation, and evaluation—rather than a linear or cyclical sequence. Such approaches emphasize the importance of understanding and leveraging the interactions within and between these processes and stakeholders to optimize the creation and movement of knowledge. By acknowledging the complex, adaptive nature of healthcare systems, complexity science advocates for continuous stakeholder engagement, transdisciplinary collaboration, and flexible strategies to effectively translate research into practice.[38]
Complexity and biology
Complexity science has been applied to living organisms, and in particular to biological systems. Within the emerging field of fractal physiology, bodily signals, such as heart rate or brain activity, are characterized using entropy or fractal indices. The goal is often to assess the state and the health of the underlying system, and diagnose potential disorders and illnesses.[citation needed]
Complexity and chaos theory
Complex systems theory is related to 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.[39] 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 the relevant equations describing the chaotic system's behavior, one can theoretically make perfectly accurate predictions of the system, though in practice this is impossible to do with arbitrary accuracy.
The emergence of complex systems theory shows a domain between deterministic order and randomness which is complex.[40] This is referred to as the "edge of chaos".[41]
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,[42] 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.[40] For recent examples in economics and business see Stoop et al.[43] who discussed Android's market position, Orlando[44] who explained the corporate dynamics in terms of mutual synchronization and chaos regularization of bursts in a group of chaotically bursting cells and Orlando et al.[45] who modelled financial data (Financial Stress Index, swap and equity, emerging and developed, corporate and government, short and long maturity) with a low-dimensional deterministic model.
Therefore, the main difference between chaotic systems and complex systems is their history.[46] Chaotic systems do not rely on their history as complex ones do. Chaotic behavior pushes a system in equilibrium into chaotic order, which means, in other words, out of what we traditionally define as 'order'.[clarification needed] 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".[47] 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 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.
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.[48][49] For example, the Internet can be represented as a network composed of nodes (computers) and links (direct connections between computers). Other examples of complex networks include social networks, financial institution interdependencies,[50] airline networks,[51] and biological networks.
^Ladyman, James; Lambert, James; Wiesner, Karoline (2013). "What is a complex system?"(PDF). European Journal for Philosophy of Science. 3: 33–67. doi:10.1007/s13194-012-0056-8. Retrieved 28 July 2024. [S]pecial issue of Science on 'Complex Systems' featuring many key figures in the field (Science 2 April 1999) [...] [:] 6. 'A complex system is literally one in which there are multiple interactions between many different components.' (40, p. 105)
^Ladyman, James; Lambert, James; Wiesner, Karoline (2013). "What is a complex system?"(PDF). European Journal for Philosophy of Science. 3: 33–67. doi:10.1007/s13194-012-0056-8. Retrieved 28 July 2024. The following quotations (apart from the last one) come from a special issue of Science on 'Complex Systems' featuring many key figures in the field (Science 2 April 1999) [:] 8. 'In recent years the scientific community has coined the rubric 'complex system' to describe phenomena, structure, aggregates, organisms, or problems that share some common theme: (i) They are inherently complicated or intricate ...; (ii) they are rarely completely deterministic; (iii) mathematical models of the system are usually complex and involve non-linear, ill-posed, or chaotic behavior; (iv) the systems are predisposed to unexpected outcomes (so-called emergent behaviour).' (14, p. 410)
^Parker, B. R. (2013). Chaos in the Cosmos: the Stunning Complexity of the Universe. Springer.
^Bekenstein, J. D. (2003). Information in the holographic universe. Scientific American, 289(2), 58-65.
^Ladyman, James; Lambert, James; Wiesner, Karoline (2013). "What is a complex system?"(PDF). European Journal for Philosophy of Science. 3: 33–67. doi:10.1007/s13194-012-0056-8. Retrieved 28 July 2024. The following quotations (apart from the last one) come from a special issue of Science on 'Complex Systems' featuring many key figures in the field (Science 2 April 1999) [...] [:] 3. 'In a general sense, the adjective 'complex' describes a system or component that by design or function or both is difficult to understand and verify. ...complexity is determined by such factors as the number of components and the intricacy of the interfaces between them, the number and intricacy of conditional branches, the degree of nesting, and the types of data structures.'(50, p. 92)
^Ladyman, James; Lambert, James; Wiesner, Karoline (2013). "What is a complex system?"(PDF). European Journal for Philosophy of Science. 3: 33–67. doi:10.1007/s13194-012-0056-8. Retrieved 28 July 2024. The following quotations (apart from the last one) come from a special issue of Science on 'Complex Systems' featuring many key figures in the field (Science 2 April 1999) [...] [:] 4. 'Complexity theory indicates that large populations of units can self-organize into aggregations that generate pattern, store information, and engage in collective decision-making.' (39, p. 99)
^Holland, John H. (2014). Complexity: A Very Short Introduction. Oxford University Press. ISBN978-0-19-966254-8. All CAS agents, whatever their particularities, have three levels of activity: 1. Performance (moment-by-moment capabilities) 2. Credit-assignment (rating the usefulness of available capabilities) 3. Rule-discovery (generating new capabilities).
^Pokrovskii, Vladimir (2021). Thermodynamics of Complex Systems: Principles and applications. IOP Publishing, Bristol, UK. Bibcode:2020tcsp.book.....P.
