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. Systems that are "complex" have distinct properties that arise from these relationships, such as nonlinearity, emergence, spontaneous order, adaptation, and feedback loops,
among others. Because such systems appear in a wide variety of fields,
the commonalities among them have become the topic of their own
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
the links their interactions.
Overview
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.
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.
As an interdisciplinary domain, complex systems draws contributions from many different fields, such as the study of self-organization from physics, that of spontaneous order from the social sciences, chaos from mathematics, adaptation from biology, and many others. Complex systems is therefore often used as a broad term encompassing a research approach to problems in many diverse disciplines, including statistical physics, information theory, nonlinear dynamics, anthropology, computer science, meteorology, sociology, economics, psychology, and biology.
Key concepts
Systems
Open systems have input and output flows, representing exchanges of matter, energy or information with their surroundings.
Complex systems is chiefly concerned with the behaviors and properties of systems.
A system, broadly defined, is a set of entities that, through their
interactions, relationships, or dependencies, form a unified whole. It
is always defined in terms of its boundary, which determines the
entities that are or are not part of the system. Entities lying outside
the system then become part of the system's environment.
A system can exhibit properties that produce behaviors which are distinct from the properties and behaviors of its parts; these system-wide or global
properties and behaviors are characteristics of how the system
interacts with or appears to its environment, or of how its parts behave
(say, in response to external stimuli) by virtue of being within the
system. The notion of behavior implies that the study of systems is also concerned with processes that take place over time (or, in mathematics, some other phase space parameterization). Because of their broad, interdisciplinary applicability, systems concepts play a central role in complex systems.
As a field of study, complex systems is a subset of systems theory.
General systems theory focuses similarly on the collective behaviors of
interacting entities, but it studies a much broader class of systems,
including non-complex systems where traditional reductionist approaches
may remain viable. Indeed, systems theory seeks to explore and describe all
classes of systems, and the invention of categories that are useful to
researchers across widely varying fields is one of systems theory's main
objectives.
As it relates to complex systems, systems theory contributes an
emphasis on the way relationships and dependencies between a system's
parts can determine system-wide properties. It also contributes the
interdisciplinary perspective of the study of complex systems: the
notion that shared properties link systems across disciplines,
justifying the pursuit of modeling approaches applicable to complex
systems wherever they appear. Specific concepts important to complex
systems, such as emergence, feedback loops, and adaptation, also
originate in systems theory.
Complexity
Systems
exhibit complexity means that their behaviors cannot be easily implied
from the very properties that make them difficult to model, and the
complex behaviors are governed entirely, or almost entirely, by the
behaviors those properties produce. Any modeling approach that ignores
such difficulties or characterizes them as noise, then, will necessarily
produce models that are neither accurate nor useful. As yet no fully
general theory of complex systems has emerged for addressing these
problems, so researchers must solve them in domain-specific contexts.
Researchers in complex systems address these problems by viewing the
chief task of modeling to be capturing, rather than reducing, the
complexity of their respective systems of interest.
While no generally accepted exact definition of complexity exists
yet, there are many archetypal examples of complexity. Systems can be
complex if, for instance, they have chaotic behavior (behavior that exhibits extreme sensitivity to initial conditions), or if they have emergent
properties (properties that are not apparent from their components in
isolation but which result from the relationships and dependencies they
form when placed together in a system), or if they are computationally
intractable to model (if they depend on a number of parameters that
grows too rapidly with respect to the size of the system).
Networks
The interacting components of a complex system form a network, which is a collection of discrete objects and relationships between them, usually depicted as a graph of vertices connected by edges. Networks can describe the relationships between individuals within an organization, between logic gates in a circuit, between genes in gene regulatory networks, or between any other set of related entities.
Networks often describe the sources of complexity in complex
systems. Studying complex systems as networks therefore enables many
useful applications of graph theory and network science. Some complex systems, for example, are also complex networks,
which have properties such as phase transitions and power-law degree
distributions that readily lend themselves to emergent or chaotic
behavior. The fact that the number of edges in a complete graph grows quadratically
in the number of vertices sheds additional light on the source of
complexity in large networks: as a network grows, the number of
relationships between entities quickly dwarfs the number of entities in
the network.
Nonlinearity
A sample solution in the Lorenz attractor when ρ = 28, σ = 10, and β = 8/3
Complex systems often have nonlinear behavior, meaning they may
respond in different ways to the same input depending on their state or
context. In mathematics and physics,
nonlinearity describes systems in which a change in the size of the
input does not produce a proportional change in the size of the output.
For a given change in input, such systems may yield significantly
greater than or less than proportional changes in output, or even no
output at all, depending on the current state of the system or its
parameter values.
Of particular interest to complex systems are nonlinear dynamical systems, which are systems of differential equations that have one or more nonlinear terms. Some nonlinear dynamical systems, such as the Lorenz system, can produce a mathematical phenomenon known as chaos. Chaos as it applies to complex systems refers to the sensitive dependence on initial conditions, or "butterfly effect,"
that a complex system can exhibit. In such a system, small changes to
initial conditions can lead to dramatically different outcomes. Chaotic
behavior can therefore be extremely hard to model numerically, because
small rounding errors at an intermediate stage of computation can cause
the model to generate completely inaccurate output. Furthermore, if a
complex system returns to a state similar to one it held previously, it
may behave completely differently in response to exactly the same
stimuli, so chaos also poses challenges for extrapolating from past
experience.
Emergence
Another common feature of complex systems is the presence of emergent
behaviors and properties: these are traits of a system which are not
apparent from its components in isolation but which result from the
interactions, dependencies, or relationships they form when placed
together in a system. Emergence
broadly describes the appearance of such behaviors and properties, and
has applications to systems studied in both the social and physical
sciences. While emergence is often used to refer only to the appearance
of unplanned organized behavior in a complex system, emergence can also
refer to the breakdown of organization; it describes any phenomena which
are difficult or even impossible to predict from the smaller entities
that make up the system.
One example of complex system whose emergent properties have been studied extensively is cellular automata.
In a cellular automaton, a grid of cells, each having one of finitely
many states, evolves over time according to a simple set of rules. These
rules guide the "interactions" of each cell with its neighbors.
Although the rules are only defined locally, they have been shown
capable of producing globally interesting behavior, for example in Conway's Game of Life.
Spontaneous order and self-organization
When emergence describes the appearance of unplanned order, it is spontaneous order (in the social sciences) or self-organization (in physical sciences). Spontaneous order can be seen in herd behavior,
whereby a group of individuals coordinates their actions without
centralized planning. Self-organization can be seen in the global
symmetry of certain crystals, for instance the apparent radial symmetry of snowflakes, which arises from purely local attractive and repulsive forces both between water molecules and between water molecules and their surrounding environment.
Adaptation
Complex adaptive systems are special cases of complex systems that are adaptive in that they have the capacity to change and learn from experience. Examples of complex adaptive systems include the stock market, social insect and ant colonies, the biosphere and the ecosystem, the brain and the immune system, the cell and the developing embryo, the cities, manufacturing businesses and any human social group-based endeavor in a cultural and social system such as political parties or communities.
Features
Complex systems may have the following features:
- Cascading failures
- Due to the strong coupling between components in complex systems, a failure in one or more components can lead to cascading failures which may have catastrophic consequences on the functioning of the system. Localized attack may lead to cascading failures and abrupt collapse in spatial networks.
- Complex systems may be open
- Complex systems are usually open systems — that is, they exist in a thermodynamic gradient and dissipate energy. In other words, complex systems are frequently far from energetic equilibrium: but despite this flux, there may be pattern stability, see synergetics.
- Complex systems may have a memory
- The history of a complex system may be important. Because complex systems are dynamical systems they change over time, and prior states may have an influence on present states. More formally, complex systems often exhibit spontaneous failures and recovery as well as hysteresis. Interacting systems may have complex hysteresis of many transitions.
- Complex systems may be nested
- 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.
- Dynamic network of multiplicity
- As well as coupling rules, the dynamic network of a complex system is important. Small-world or scale-free networks 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, the termites in a mound have physiology, biochemistry and biological development that are at one level of analysis, but their social behavior and mound building is a property that emerges from the collection of termites and needs to be analysed 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, effect is always directly proportional to cause.
- Relationships contain feedback loops
- Both negative (damping) and positive (amplifying) feedback are always found in complex systems. The effects of an element's behaviour are fed back to in such a way that the element itself is altered.
History
A perspective on the development of complexity science: http://www.art-sciencefactory.com/complexity-map_feb09.html
Although it is arguable that humans have been studying complex
systems for thousands of years, the modern scientific study of complex
systems is relatively young in comparison to established fields of
science such as physics and chemistry. The history of the scientific study of these systems follows several different research trends.
In the area of mathematics, arguably the largest contribution to the study of complex systems was the discovery of chaos in deterministic systems, a feature of certain dynamical systems that is strongly related to nonlinearity. The study of neural networks was also integral in advancing the mathematics needed to study complex systems.
The notion of self-organizing systems is tied with work in nonequilibrium thermodynamics, including that pioneered by chemist and Nobel laureate Ilya Prigogine in his study of dissipative structures. Even older is the work by Hartree-Fock c.s. on the quantum-chemistry
equations and later calculations of the structure of molecules which
can be regarded as one of the earliest examples of emergence and
emergent wholes in science.
One complex system containing humans is the classical political economy of the Scottish Enlightenment, later developed by the Austrian school of economics, which argues that order in market systems is spontaneous (or emergent) in that it is the result of human action, but not the execution of any human design.
Upon this the Austrian school developed from the 19th to the early 20th century the economic calculation problem, along with the concept of dispersed knowledge, which were to fuel debates against the then-dominant Keynesian economics. This debate would notably lead economists, politicians and other parties to explore the question of computational complexity.
A pioneer in the field, and inspired by Karl Popper's and Warren Weaver's works, Nobel prize economist and philosopher Friedrich Hayek dedicated much of his work, from early to the late 20th century, to the study of complex phenomena, not constraining his work to human economies but venturing into other fields such as psychology, biology and cybernetics. Gregory Bateson
played a key role in establishing the connection between anthropology
and systems theory; he recognized that the interactive parts of cultures
function much like ecosystems.
While the explicit study of complex systems dates at least to the 1970s, the first research institute focused on complex systems, the Santa Fe Institute, was founded in 1984. 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. Today, there are over 50 institutes and research centers focusing on complex systems. A scientific society called Complex Systems Society organizes every year a general conference on these topics.
Applications
Complexity in practice
The
traditional approach to dealing with complexity is to reduce or
constrain it. Typically, this involves compartmentalisation: dividing a
large system into separate parts. Organizations, for instance, divide
their work into departments that each deal with separate issues.
Engineering systems are often designed using modular components.
However, modular designs become susceptible to failure when issues arise
that bridge the divisions.
Complexity management
As projects and acquisitions
become increasingly complex, companies and governments are challenged
to find effective ways to manage mega-acquisitions such as the Army Future Combat Systems. Acquisitions such as the FCS
rely on a web of interrelated parts which interact unpredictably. As
acquisitions become more network-centric and complex, businesses will be
forced to find ways to manage complexity while governments will be
challenged to provide effective governance to ensure flexibility and
resiliency.
Complexity economics
Over the last decades, within the emerging field of complexity economics new predictive tools have been developed to explain economic growth. Such is the case with the models built by the Santa Fe Institute in 1989 and the more recent economic complexity index (ECI), introduced by the MIT physicist Cesar A. Hidalgo and the Harvard economist Ricardo Hausmann. Based on the ECI, Hausmann, Hidalgo and their team of The Observatory of Economic Complexity have produced GDP forecasts for the year 2020.
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".
Complexity and modeling
One
of Friedrich Hayek's main contributions to early complexity theory is
his distinction between the human capacity to predict the behaviour of
simple systems and its capacity to predict the behaviour of complex
systems through modeling.
He believed that economics and the sciences of complex phenomena in
general, which in his view included biology, psychology, and so on,
could not be modeled after the sciences that deal with essentially
simple phenomena like physics.
Hayek would notably explain that complex phenomena, through modeling,
can only allow pattern predictions, compared with the precise
predictions that can be made out of non-complex phenomena.
Complexity and chaos theory
Complexity theory is rooted in chaos theory, which in turn has its origins more than a century ago in the work of the French mathematician Henri Poincaré. Chaos is sometimes viewed as extremely complicated information, rather than as an absence of order.
Chaotic systems remain deterministic, though their long-term behavior
can be difficult to predict with any accuracy. With perfect knowledge of
the initial conditions and of the relevant equations describing the
chaotic system's behavior, one can theoretically make perfectly accurate
predictions about the future of the system, though in practice this is
impossible to do with arbitrary accuracy. Ilya Prigogine argued that complexity is non-deterministic, and gives no way whatsoever to precisely predict the future.
The emergence of complexity theory shows a domain between deterministic order and randomness which is complex. This is referred as the "edge of chaos".
A plot of the Lorenz attractor.
When one analyzes complex systems, sensitivity to initial conditions,
for example, is not an issue as important as it is within chaos theory,
in which it prevails. As stated by Colander,
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.
Therefore, the main difference between chaotic systems and complex systems is their history.
Chaotic systems do not rely on their history as complex ones do.
Chaotic behaviour pushes a system in equilibrium into chaotic order,
which means, in other words, out of what we traditionally define as
'order'.
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".
In a sense chaotic systems can be regarded as a subset of complex
systems distinguished precisely by this absence of historical
dependence. Many real complex systems are, in practice and over long but
finite time periods, robust. However, they do possess the potential for
radical qualitative change of kind whilst retaining systemic integrity.
Metamorphosis serves as perhaps more than a metaphor for such
transformations.
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. For example, the INTERNET can be represented as a network composed of
nodes (computers) and links (direct connections between computers). Its
resilience to failures was studied using percolation theory.
Other examples are social networks, airline networks, biological networks and climate networks.
Networks can also fail and recover spontaneously. For modeling this phenomenon see Majdandzic et al.
Interacting complex systems can be modeled as networks of networks. For their breakdown and recovery properties see Gao et al.
Traffic in a city can be represented as a network. The weighted links
represent the velocity between two junctions (nodes). This approach was
found useful to characterize the global traffic efficiency in a city. The complex pattern of exposures between financial institutions has been shown to trigger financial instability.
General form of complexity computation
The computational law of reachable optimality
is established as a general form of computation for ordered systems and
it reveals complexity computation is a compound computation of optimal
choice and optimality driven reaching pattern over time underlying a
specific and any experience path of ordered system within the general
limitation of system integrity.
The computational law of reachable optimality has four key components as described below.
1. Reachability of Optimality: Any intended optimality
shall be reachable. Unreachable optimality has no meaning for a member
in the ordered system and even for the ordered system itself.
2. Prevailing and Consistency: Maximizing reachability to
explore best available optimality is the prevailing computation logic
for all members in the ordered system and is accommodated by the ordered
system.
3. Conditionality: Realizable tradeoff between
reachability and optimality depends primarily upon the initial bet
capacity and how the bet capacity evolves along with the payoff table
update path triggered by bet behavior and empowered by the underlying
law of reward and punishment. Precisely, it is a sequence of conditional
events where the next event happens upon reached status quo from
experience path.
4. Robustness: The more challenge a reachable optimality can accommodate, the more robust it is in term of path integrity.
There are also four computation features in the law of reachable optimality.
1. Optimal Choice: Computation in realizing Optimal Choice
can be very simple or very complex. A simple rule in Optimal Choice is
to accept whatever is reached, Reward As You Go (RAYG). A Reachable
Optimality computation reduces into optimizing reachability when RAYG is
adopted. The Optimal Choice computation can be more complex when
multiple NE strategies present in a reached game.
2. Initial Status: Computation is assumed to start at an
interested beginning even the absolute beginning of an ordered system in
nature may not and need not present. An assumed neutral Initial Status
facilitates an artificial or a simulating computation and is not
expected to change the prevalence of any findings.
3. Territory: An ordered system shall have a territory
where the universal computation sponsored by the system will produce an
optimal solution still within the territory.
4. Reaching Pattern: The forms of Reaching Pattern in the
computation space, or the Optimality Driven Reaching Pattern in the
computation space, primarily depend upon the nature and dimensions of
measure space underlying a computation space and the law of punishment
and reward underlying the realized experience path of reaching. There
are five basic forms of experience path we are interested in,
persistently positive reinforcement experience path, persistently negative reinforcement experience path, mixed persistent pattern experience path, decaying scale experience path and selection experience path.
The compound computation in selection experience path includes
current and lagging interaction, dynamic topological transformation and
implies both invariance and variance characteristics in an ordered
system's experience path.
In addition, the computation law of reachable optimality gives
out the boundary between complexity model, chaotic model and
determination model. When RAYG is the Optimal Choice computation, and
the reaching pattern is a persistently positive experience path,
persistently negative experience path, or mixed persistent pattern
experience path, the underlying computation shall be a simple system
computation adopting determination rules. If the reaching pattern has no
persistent pattern experienced in RAYG regime, the underlying
computation hints there is a chaotic system. When the optimal choice
computation involves non-RAYG computation, it's a complexity computation
driving the compound effect.
