Complex Systems

  • Hajime KitaEmail author
Living reference work entry


What are complex systems? One answer may be systems having complex configurations. Then, where such complexity coming from? Another answer may be systems that show complex behaviors while their configurations are rather simple. Then, what are the underlying mechanisms of such behaviors? This chapter gives an overview from various aspects such as emergence of complex behavior in physical systems, biological and ecological systems as complex systems, artificial systems developed by human, and social systems as complex systems. While some complex behaviors can be observed in rather simple physical systems and physical and mathematical understanding give guiding principles of emergence of such complexity, biological and artificial systems are far from it. In this chapter, the author tries to give overview from a viewpoint of self-reproductive systems and adaptive behaviors of them. As a possible scenario, once a system acquires self-reproductive abilities, i.e., to make copies of the system by itself, adaptation in the sense of performance of self-reproduction starts. Then various adaptive mechanisms would also emerge on the base of the self-reproductive system. In such adaptive mechanisms, information processing plays intrinsic roles both in biological and artificial systems, and we discuss complex systems also from this point of view.


Complexity Self-reproduction Adaptation Evolution Biological systems Artificial systems 


Complex systems, or complexity of systems, are difficult words to explain clearly. Intuitively, we are surrounded by many complex systems, e.g., our body, as well as other species, consisting of large number of cells of various types. Recent technologies have enabled construction of large-scale artificial systems such as space crafts, power systems, information systems, etc. We are now living in very complex socioeconomic systems. However, if we try to capture complexity of systems formally, we encounter the problem of how to define or to measure the complexity of the systems. (Mitchell listed various viewpoints to measure complexity of systems in Mitchell (2009)) For example, if we try to describe some amount of gas consisting of molecules, very large description is needed because each of the molecules in the gas moves randomly. However, the gas as a whole follows a very simple equation among the temperature, volume, and pressure, and hence from a macroscopic point of view, the gas is rather a simple system.

Complex systems around us have two natures. That is, the system has some orders far from the complete randomness or disorder on one hand, and on the other hand, the orders are not simple. All the existing concrete systems have to follow physical principles. Then how complexities of systems arise under the constraints of physical principles? One scenario of emergence of complex systems is adaptation of the systems. That is, systems have some adaptive mechanisms, and through accumulation of information obtained by adaptation, some orders of the systems emerge as the results of adaptation. In this chapter, we make an overview of complex systems in physical, biological, and artificial systems from the viewpoint of accumulation of information.

Physical Aspect of Complex Systems

Symmetry Breaking in Thermal Equilibrium

In this chapter, we focus on macroscopic systems consisting of many elements such as atoms or molecules. If a system consisting of atoms or molecules is isolated, i.e., no energy and material flow exists between the system and its surroundings, it has to obey the second law of thermodynamics. That is, the total entropy can never decrease over time. In the end, the system has to stay at the thermal equilibrium of the maximum entropy. For example, if we put some amount of hot waters and cold water thermally connected, the temperatures of both get closer, and finally it stays at the same intermediate temperature as a thermal equilibrium, and it never be back to hot and cold. That is, thermal equilibrium is the state that loses orders mostly.

However, even in the thermal equilibrium, we observe several different phases depending on the temperature. For example, water takes form of steam (gas) in high temperature, and by cooling it down, it becomes liquid and then ice (solid). Such changes are called “phase transition.” It is a collective behavior of material consisting of very large number of molecules, and it occurs in balance of thermal motion/vibration of each molecule and forces among the molecules. In high temperature, the effect of the force is small compared with thermal energy of the molecules, and each molecules can move randomly as gas. Contrary to this, if temperature gets low, forces between molecules constrain the movement of molecules, and they are crystallized.

An interesting example of phase transition is a magnet. If we heat a magnet, at a some critical temperature, magnetization is lost. If we cool it down in external magnetic field, it shows spontaneous magnetization, and we can turn its polarity by changing the external magnetic field.

Emergence of such phenomenon can be understood by statistical mechanics with “Ising model.” Microscopically, a magnet consists of elements having magnetic polarity, i.e., micro magnets. In the Ising model, such micro magnets (or Ising spins) are modeled as Si ∈ {−1, 1} having down (–1) or up (1) polarity, and the total magnetization is expressed by (Notation of the model follows a Japanese article by Kabashima (2007))
$$ M=\frac{1}{N}\sum \limits_N\sum \limits_S{S}_iP(S) $$
where N is the number of Ising spins and P(S) is probability that the system takes configuration S = S1, ⋯SN called canonical distribution, which is given by
$$ P(S)=\frac{1}{Z}\exp \left(\frac{-H(S)}{T}\right) $$
where Z is a coefficient for normalization given by \( Z={\sum}_S\exp \left(-\frac{H(S)}{T}\right) \) and T is temperature. H(S) is the energy function given by
$$ H(S)=-\sum \limits_{<i,j>}{J}_{ij}{S}_i{S}_j-\sum \limits_i{h}_i{S}_i $$
where Jij is constants that express mutual interaction among the micro magnets and hi is an external magnetic field to the small magnet Si, and <i, j> in summation means configuration of interacting spins such as nearest neighbor. The canonical distribution means the system takes every configuration with almost equal probability in very high temperature and takes only the minimums near-zero temperature. The material that shows spontaneous magnetization (ferromagnetic material) is modeled as Ji. j ≥ 0, and then the energy function with no external field takes its minimum when all the micro magnets take same values such as “all up (1)” or “all down (–1).” While the P(S) is symmetric for these two states, theoretical analysis shows that the system takes only one of the minimums, i.e., symmetry breaking occurs, and it explains the spontaneous magnetization in low temperature. Analysis also explains that magnetization is lost at a some temperature. Ising model is proposed as the model for magnetization, phenomenon under rather random interaction among Si is studied as “spin glass,” whose energy function has various local minimums.

This sort of symmetry breaking emergently occurs due to interaction of many elements and is important for adaptive system because it can be used as a memory so as to accumulate the information obtained in adaptation. Anderson discussed such emergence as “More is Different” (Anderson 1972).

Emergence of Patterns in Non-equilibrium Systems

In the thermal equilibrium, phenomena such as phase transition and symmetry breaking appear; however, it still obeys the second law of thermodynamics. That is, macroscopic spatial and temporal patterns will disappear along evolution to equilibrium.

However, in non-equilibrium systems, self-organization of macroscopic spatial and temporal patterns can be observed. Thermal equilibrium is assumed that the system is isolated, but non-equilibrium systems have some interaction with its surrounding such as energy flow. Because of such openness, it can abandon the entropy (disorder) yielded in the system to outside and can sustain its order. These sorts of pattern formation are named “dissipative structure” by Nicolis and Prigogine (1977).

Typical examples are Bénard cells and Belousov-Zhabotinsky reaction. The Bénard cells are cells of eddy current observed in liquid in a flat vessel heated at bottom and cooled at the top. In such situation, when difference of the temperatures at the both sides is small, heat is transferred by conduction. It is a state continuous to the thermal equilibrium. However, beyond some temperature, conduction gets unstable, and molecules of liquid form cells of eddy currents named Bénard cells. It is brought about by the motion of liquid that is heated at the bottom, gets light, and goes up. At the top, the liquid is cooled and gets heavy and goes down.

Belousov-Zhabotinsky reaction is an interesting example that spatial and temporal pattern (i.e., oscillation) occurs in a chemical reaction. This response is brought about by the two chemical reactions sharing same material for reaction and hence interacting each other.

Mathematical Aspect of Complex Systems

As well as physical principles, mathematical understanding is also important in discussion of complex systems. It should be noted that the physical principle gives basis of existence of actual systems, but it can be applied only to appropriate physical configurations. Contrary to this, applicability of mathematical theory is not limited to particular physical configuration. They often applied to quite different systems such as physical, biological, and social systems that share a same mathematical structure. For example, the aforesaid Ising model has been also studied as a model for neural networks and associative memories (Haykin 1994).

While there are various mathematical topics relating to complex systems, this section focuses on the study of nonlinear dynamical systems of continuous variables.

Nonlinear Dynamical Systems

For description of dynamics of systems having continuous state, we often use differential equation
$$ \frac{\mathrm{d}x(t)}{\mathrm{d}t}=F\left(x(t),u(t)\right) $$
where x(t) is a state vector of the system and u(t) is an external force vector.
Linear differential equation is given by
$$ \frac{\mathrm{d}x(t)}{\mathrm{d}t}= Ax(t)+ Bu(t) $$
where A and B are matrices of appropriate dimensions. In this equation, the RHS is given by a linear function of the state variables. This type of equation can be solved analytically. Its autonomous response, i.e., solution of the equation without external force term
$$ \frac{\mathrm{d}x(t)}{\mathrm{d}t}= Ax(t) $$
is limited to combination of terms of product of exponential functions, triangular functions, and power functions, and hence the state variables only go to 0 and infinity or keeps oscillation.

If a system has dynamics of a nonlinear type, i.e., terms consisting of products of state variables appear in RHS, various interesting behaviors are observed.

In the dynamical system, evolution of the state variable x(t) given initial values x(0) is called an “orbit.” One of the interests in dynamical system is where the orbit goes. The set of points which is invariant along evolution and absorb other orbits is called an “attractor” of the dynamical system. In nonlinear dynamical systems, we see variation of attractors such as multiple sinks, limit cycles, and strange (or chaos) attractors.

Oscillation and Synchronization

Stable oscillation can be understood as absorption to an attractor named a limit cycle, a closed curve on the state space on which state variables evolve repetitively in the nonlinear dynamics. Further, we observe in connected oscillating systems that oscillations share a same frequency even if the separated subsystems oscillate in different frequencies. Such phenomenon is called “synchronization.” Synchronization can be seen in various living systems and also applied to artificial systems (Strogatz 2003; Pikowsky et al. 2001). Synchronization provides efficient ways to achieve a coherent motion in a system of large degree of freedoms with a rather simple configuration without, e.g., a central controller, and hence it can be a powerful tool in emergence of complex systems.

The following is a model proposed by Kuramoto (1984) and Pikowsky et al. (2001) for study of synchronization in globally coupled oscillators
$$ \frac{d{\phi}_k}{dt}={\omega}_k+\frac{\varepsilon }{N}\sum \limits_{j=1}^N\sin \left({\phi}_j-{\phi}_k\right) $$
where ϕk and ωk are phase and natural angular velocity of oscillator k, respectively. The strength of coupling is represented by the parameter ε.

It is shown that there exists a phase transition of synchronization among the oscillators in the limit of N → ∞. That is, there is a critical value in the parameter ε, and beyond this value, synchronization occurs among some oscillators.


Another interesting phenomenon is chaos (Strogatz 2015). In chaos dynamics, system stays in a limited area but go around in a very complicated attractor named a “strange attractor.” Chaos dynamics has both stability that orbits are absorbed to the attractor and instability that small difference of states is enlarged in future on the attractor.

Depending on values of parameters, behavior such as synchronization and chaos appears or disappears. Such change is called “bifurcation.” Phase transitions in physical systems are closely related to bifurcation in nonlinear dynamical systems.

Biological and Ecological Systems as Complex Systems


We see pattern formation in non-equilibrium system beyond the limit of thermal equilibrium. However, it still stays rather simple patterns. On the other hand, we observe further complexity in biological systems. One of the salient differences of living systems and nonliving systems is self-reproduction. That is, a cell, a basic unit of life, has the ability of making its copies and increasing in number. Thus, the cells sustain themselves as population against destroying of cells due to several reasons.

Self-reproduction of the living system is achieved by genetics. That is, configuration of a cell is coded as arrays of four types of nucleotide on DNA in the cell. By decoding the nucleotide sequence on DNA into protein, i.e., a sequence of amino acids, the cell achieved their functions as a living system. One of such functions is that of copying its own, and the most key element is copying of DNA. From a viewpoint of information processing, it is an interesting symbolic process because the DNA plays two roles of “program” and “data.” As a program, information on the DNA is decoded and executed as cell functions including the function of copying itself. At the same time, it is treated as data to be copied in self-reproduction process. It is quite similar to the stored-program architecture commonly used in modern digital computers. In computers, program is expressed as an instruction sequence on the memory. In loading of the program, it is treated as data just to transfer from a disc drive to the memory by another program (usually by an operating system), once it is loaded on the memory, then it is treated as program, and by decoding the instruction in the program, it is executed.

Formal and computational study of self-reproduction was carried out with cell automata, a symbolic system that a cell arrayed in a grid. Each cell changes its state among a finite set depending on the states of surrounding cells including itself. It was first studied by von Neumann (1966), (He is also known as a writer of a report proposing aforesaid stored program architecture of digital computers (von Neumann 1945). His work on self-reproduction on cell automata was published after his death.) and later various sorts of models are studied (Sipper 1998).

Dynamics of Self-Reproductive Systems

If a self-reproductive system starts reproduction in a constant speed, the number of the systems increases exponentially, i.e.,
$$ 1\to 2\to 4\to \cdots {2}^N\cdots $$
However, the systems encounter some environmental limit to sustain them. The following logistic equation (Notation of the mathematical models in this section follows those in Hofbauer and Sigmund (1998)) is a model of such self-reproductive process
$$ \frac{\mathrm{d}x}{\mathrm{d}t}= rx\left(1-\frac{x}{K}\right) $$
where x is population size of self-reproductive systems and r is a parameter of reproduction speed in the case environmental limit has no effect. The parameter K expresses a capacity of environment. If population size x comes near to this limit, growth slows down. This equation can be solved analytically as (Different from the continuous time equation in Eq. (9), a discrete-time version of this model x(t + 1) = ax(t)(x(t) − 1) shows complex behavior depending on the parameter a such as doubling bifurcation going to chaos).
$$ x(t)=\frac{Kx(0){e}^{rt}}{K+x(0)\left({e}^{rt}-1\right)} $$
Coexisting situation of several types of self-reproduction systems mutually relating, i.e., an ecological system of mutually interacting species, can be discussed by extending this equation. One of famous models is Lotka-Volterra equations of predator-prey relationship given by
$$ \frac{\mathrm{d}x}{\mathrm{d}t}=x\left(a- by\right) $$
$$ \frac{\mathrm{d}y}{\mathrm{d}t}=y\left(-c+ dx\right) $$
where x and y are populations of prey and predator, respectively, and a, b, c, and d are positive parameters. This model shows oscillating behavior of the populations of the two species.
Further, more general extension to treat ecological systems is discussed with the following replicator dynamics:
$$ \frac{\mathrm{d}{x}_i}{\mathrm{d}t}={x}_i\left[{f}_i(x)-\overline{f}(x)\right],\mathrm{for}\ x=1,\dots, n $$
$$ \overline{f}(x)=\sum \limits_{i=1}^n{x}_i{f}_i(x) $$
where xi is the proportion of type i in the whole population, x is a vector whose i-th component is xi, fi is fitness of type i, and \( \overline{f} \) is the average fitness of the population.

A study framework combining game theoretical setting among competing species as interacting payoff functions with reproduction dynamics such as replicator equations is called evolutionary games. With this framework, stability concept of the game called “the evolutionary stable strategies (ESS)” is proposed. ESS is defined as a state that existing population excludes intrusion of any other species because of their superiority in payoff over those of intruders.

While these models explain the dynamics of mutually related self-reproducing species, they only treat behaviors of population of species of known configurations. To discuss adaptive behavior and change of each self-reproducing system, we have to include the concept of evolution.

Evolution of Self-Reproducing Systems

In natural science, physical principles are given in forms of equations that the natural system follows, and therefore adaptation is difficult to settle in its concept. However, with the context of self-reproducing system, the concept of “adaptation” can be introduced. That is, if a self-reproducing system utilizes its environment well through adaptation, its reproduction process becomes more stabilized against the attack of increasing entropy and accelerates its reproduction. Adaptation of self-producing systems will be achieved with the following two processes:
  • To generate novel configurations of the self-reproducing systems

  • To evaluate the novel configurations as well as known ones from the viewpoint of their performance in reproduction

Remembering the symbolic representation of genetic information, the following can be one way for adaptation:
  • Give some perturbation to the genetic information in reproduction process, e.g., failure in coping of DNA. Then, through decoding it, it can change the performance of the reproduced system.

  • Above perturbation occurs at an individual system level. If perturbed individual system performs well in production of its offspring, it is an actual evaluation of the novel configuration.

It is a basic concept of “natural selection theory of evolution” or Darwinism. Information in gene (genotype) is interpreted to actual configuration of lives (phenotype), and as a result, it decides fitness or survivability of the offspring. It should be noted that the information flow is one way from the genotype to the phenotype, and success of individual without genetic variation is not fed back to genotype. While the above process is individual based, many species takes sexual reproduction. That is, each individual has two sets of gene (diploid) and a couple of individuals reproducing their offspring by inheriting half (haploid) of each parent (On sexual reproduction, there have been proposed various discussions both from the viewpoints of benefit and loss in self-reproduction (Maynard Smith and Szhtháry 1995)).

Genetic algorithms (GA) are computational method imitating the evolution process of sexual reproduction. In GA, solution of some optimization problem is coded as an individual consisting of a sequence of symbols (Usually, each individual possesses one set of solution. That is, diploidy is not adopted as a common method.). Starting with population of individuals randomly generated, the following steps are applied repeatedly:
  1. 1.

    Selection/reproduction: According to the performance as a solution of the optimization problem, each individual is evaluated, and better individuals produce their copies more, and poor individuals are excluded.

  2. 2.

    Crossover: Make couples of individuals in the population and exchange some part of their symbol sequences among each couple.

  3. 3.

    Mutation: Give random perturbation to the individual’s symbol sequence.


Starting with pioneer works by Holland (1975) and Goldberg (1989), GA has been studied intensively and applied to various actual problems (Mitchell 1996). There are also proposed various computational methods inspired by evolution including GA; they are called evolutionary computation.

Behind such study, there are computational difficulties of combinatorial explosion. Assuming an optimization problem of decision variables each takes a value in a finite set, e.g., 0 or 1, possible configurations (or size of search space) are given by combinations of values of variables, and they grow exponentially according to the number of decision variables. While some problems can be solved efficiently with adequate algorithms, many interesting problems belong to the problem class named NP-hard, and to find efficient algorithms (theoretically, algorithms terminated in polynomial time of the problem size) is expected difficult, and some heuristic methods such as GA that provides good solutions within adequate computation time are needed. Studies on GAs show the crossover operation can be an effective search method. However, genetic diversity in parental population is intrinsic in searching with the crossover, and selection/reproduction should be carried out considering maintenance of genetic diversity.

As well as applications of GA to engineering problems, GA has been also used for a method to study complex adaptive systems (Mitchell 1996). For such study, abstract fitness landscape model which has some computational difficulties as well as simpleness as an example is required. NK model proposed by Kauffman and Levin is often used for such purposes (Kauffman and Levin 1987).

In engineering applications of GA, fitness of each individual is simply evaluated as performance in the problem separately. In study of complex systems of interacting individuals, they affect their fitness mutually, and evolution of strategies against others becomes issues. Iterated prisoners’ dilemma (IPD) (Poundstone 1992) is a framework often used. Axelrod’s study using contests for computer programs was pioneer work in computational approach in study using IPD (Axelrod 1984). Later, evolution of strategies in IPD was also studied by Axelrod (1989). It should be noted that, in study of complex systems through adaptation, actual difficulty in adaptation problem is an intrinsic issue. Experience of applying GA to actual problems revealed it, which is difficult to approach in classical methods such as models using differential equations for study on complex systems shown in “Evolution of Self-Reproducing Systems”.

Natural selection theory is the theory that can explain adaptation through evolution. However, evolution under no selection pressure needs other explanation. Neutral theory of molecular evolution (Kimura 1983) is a theory for such situation. With finite population, random change of gene which has no effect of selection pressure can remain their offspring probabilistically under the reproduction process of individual. Occasionally, such change spreads to all the population. It is known as “random genetic drift.” Theoretical analysis shows that accumulation speed of fixation of random change does not depend on the population size, and it is utilized as “molecular clock” to investigate evolution of species using their DNA sequences.

Variety of Evolution

Looking at actual lives and their ecological systems, we notice variety of results of evolution due to interaction among individuals and genetic systems. For example:
  • Sexual selection. In animals taking sexual reproduction, to select their partners, or to be selected as partners for reproduction, are important factors for survival. Occasionally, males and females of some species take behaviors to select or to be selected as partners. If such behavior is coded in gene, it will evolve via mutual selection.

  • Evolution of altruism and social behavior. While carrying over the genetic information occurs individual based, there observed altruism, i.e., to help others that increase fitness of others while its own fitness decreases. It is difficult to explain with an individual-based view of natural selection. However, it can be understood by extending fitness concept considering survival of genetic information shared by others.

  • Evolution of symbiotic relationship. Plants in Angiospermae use insects for their sexual reproduction. That is, insects carry pollen of one flower to others for pollinization. Instead, insects obtain honey as food from plants. Thus, symbiotic relationship, or win-win relationship between plants and insects, evolved via evolution of each species. In predator-prey relationship, the prey evolves not to be caught, e.g., to hide themselves in background, to detect predators’ approach, or to escape quickly from their attack on one hand. On the other hand, the predator evolves to improve their ability to catch the prey.

These sorts of evolution are beyond simple understanding of selection of individuals under selection pressure, and we have to think genetic reproduction process as systems under mutual interaction of individuals. Maynard Smith and Szhtháry gave overview of symbiosis and social aspect of living system from the viewpoint of evolution (Maynard Smith and Szhtháry 1995).

Multilevel Adaptive Systems

Actual living systems have adaptive behavior in many levels. It can be understood that the living systems were enabled to construct various adaptive mechanisms with the basic adaptive mechanism of genetics and evolutions. This picture was given by Gell-Mann (1994) and Maynard Smith and Szhtháry (1995). The following are examples of emerged complex adaptive systems in living system:
  • Differentiation of cells and morphogenesis: Multiple cell creatures not just make copy of a single cell, but it differentiates each other and makes a body of the creature that works well as a total.

  • Immune system: Some animals have immune mechanism that detects intrusion of microorganisms and destroys them inside their bodies. The immune process is very adaptive because it has to detect unknown intruders and to accelerate their response to the novel intruders on one hand, and on the other hand it responds well to previous intruders by accumulating its experience in some way.

  • Homeostasis: Some animals control their body state such as body temperature constant in changing surroundings. It is achieved by adjusting metabolism through some feedback mechanisms.

  • The nervous system and brain: The nervous system and brain achieved fast responses of animals. On the neuron, information is transferred quickly using neural spikes, and information is processed via many neurons in, e.g., a brain. It also achieved learning through adjustment of interconnection among neurons. Animals detect outside situation via sensory cells, transfer and process the sensory information via the nervous system, and changing their behavior by controlling, e.g., mussels as actuators that affect the outside.

  • Various activities taken by human: Human has various adaptive systems such as language, technology, etc. It will be discussed in the next section.

Artificial Systems as Complex Systems

Among various species in living system, human being constructed various sorts of artificial. As total, complex systems are formed as artificial.

Symbolic Processing of Information

In biological systems, genetics combined with evolution acts for adaptive self-reproduction. That is, the genetic system accumulates and transfers the information of adaptation, and it plays a fundamental role for aforesaid emergence of multilevel complex adaptive systems. Similar to this, language, a symbolic information processing system, plays basic role in emergence in artificial system as complex adaptive systems.

Origin of language is a difficult issue to study because use of oral language disappear immediately. Maynard Smith and Szhtháry (1995) gave an overview of studies on origin of language.

Acquisition of oral languages by human, followed by written languages, enables complex behavior of human being, i.e., recognizing surroundings, communication to others, recording information, thinking with language, making decision, and coordinating behaviors among people. Learning can be observed in behaviors of other animals, but in humankind, teaching or education as well as learning encourages transferring the acquired knowledge to the next generation outside the genetic systems.

At the same time, language itself evolved in increasing vocabularies to express concrete things, using it also to express abstract matters such as relationship, and establishing grammatical rules to express complex things.

Further, development of language to treat numbers, quantities, and their relationship has established mathematics. Logical and deductive inference enabled accumulation of mathematical knowledge, and it has encouraged advance of mathematics itself. Language of treating numbers, or mathematics at the same time, provided basis for development of more complex systems such as sciences and technologies as well as economical and other social systems of human society large in population size.

Modern society have increased ability of information processing with progress in technologies in various types:
  • Printing with movable type

  • Recording and duplication of images using photography and xerography

  • Electric/electronic communication such as telegraph, telephone, radio, television, and the Internet

  • Information processing with digital computers

Sciences and Technologies

Another enabler of artificial systems is invention of tools using external things for particular purposes, which modifies them to fit more to their objectives. It includes use of fire, a powerful chemical reaction to obtain heat, light, and high temperature.

Combined with ability of symbolic information processing, information for technologies are accumulated and improved in various aspects, i.e., in treating material, energy, and information with invention of machines.

Technologies are knowledge to apply natural phenomena to some purposes, and natural sciences are knowledge to understand the nature. These two activities are mutually dependent. That is, sciences require technologies to investigate the nature. Technologies also posed scientific questions of understanding phenomena already used in technologies. Through evolution of sciences, knowledge of the nature was accumulated as well as knowledge on scientific ways for establishing knowledge.

From science to technology, sciences provided technologies with knowledge of sophisticated understanding, scientific limit, and alternatives to achieve similar purposes. In modern society, sciences and technologies are tightly coupled because of sophistication of both. However, technologies are not simple applications of sciences. They also require understanding of human need and methodologies of engineering to construct and operate complex artificial systems. From the viewpoint of complex systems, scientists work for creation of new findings in the field. Their findings are evaluated by peer review from the viewpoint of novelty, consistency with the existing scientific knowledge, and consistency with the results of observation and experiments along the methodologies in sciences (As pointed out by Anderson (1972), natural sciences have been studied in hierarchy, such as elementary particle physics, multi-body physics, chemistry, molecular biology, physiology, and psychology. To establish scientific knowledge would be more difficult along this hierarchy. For example, there was a long debate between the scientists supporting the natural selection theory and those supporting neutral theory of molecular evolution (Kimura 1983)).

Then knowledge is accumulated through scientific publication, and successful activities are rewarded through allocation of resources for further study. In modern society, scientific activities are closely related to development of technologies and education, and they are supported by government and companies. As for technologies, knowledge has been proved through realization and evaluated through success in application, and further resource for development is allocated. Such proved knowledge has been accumulated as well as science. It also should be noted that in science and technologies, those standards for weight and measures, mechanical and electronic parts, etc. also play important role in reuse of the accumulated information. Arthur has discussed technologies as a whole from a viewpoint of their evolution (Arthur 2009), and he pointed out relationship between technologies and social systems such as economics as well as sciences. He also pointed out that regions or communities that can advance development of technologies such as Silicon Valley, and in such communities, they share various information that is difficult to be replicated by others referring the “tacit knowledge” proposed by Polanyi (1966).

Usage of technologies is intervention to the existing systems such as nature and human society. However, both nature and human society are complex adaptive systems by themselves, and we do not have sufficient knowledge on them. Hence intervention with technologies to them may cause unexpected results such as problems of ozone hole and global warming.

Social Systems

As well as sciences and technologies, social systems are also important artificial as complex systems. Society includes individuals, and each human itself is adaptive system having some autonomy, and hence there occur various conflicts or cooperation in the society.

Modern society include various social systems such as legal and political systems, economic systems, management systems for organizations, education systems, etc. This section gives brief overview of the social system from the viewpoint of complex systems.

Legal and political systems were evolved to govern the society through establishment of rules as written laws and force people to obey them. Economical systems including ownership, money, market, accounting, etc. enabled people to produce, exchange, and consume products in societies of large population. It also enabled various trials by people with economic freedom and accumulation of capital, which lead to innovations. In both systems, they have become complex through accumulation of information and refining rules, institutional design, and usage of technologies.

In economics as an example of study of social systems, classical theory of economics rather focused on the function of competitive market to find equilibrium prices which achieve the economically optimal allocation of resources and macroscopic circulation of economy.

However, obviously society of humankind is physically open systems far from thermal equilibrium, and behind economic circulation, there is energy and material flow from/to surroundings to abandon the entropy yielded inside the human society. At the same time, inside the human society, there occurs accumulation of information obtained in adaptation. With pioneering work by Nelson and Winters (1982), evolutionary economics (Andersen 1994) studies the economic systems from the viewpoint of its evolution. Arthur pointed out “increase return” in economic system (Arthur 1994). It means that a company that sells more obtains larger profit per product. While it is against the assumption in finding an equilibrium in the competitive market, it works as evaluation of adaptive behavior of a company and to encourage reinvestment of the profit.

As methodology for study, evolutionary economics and other study of social systems (Axelrod 1984, 1989) employ computational approaches such as agent-based and/or evolutionary simulation to attack the study of evolving systems (Deguchi 2004; Anderson 1972; Arthur 1994; Gilbert 2008) as well as mathematical analysis used in conventional economics.

Information Processing as an Engine for Emergence of Complex Systems

Requirements of Adaptation

In genetics, genetic information on DNA is transferred to the population through self-reproduction of individuals. If self-reproduction is successful, i.e., an individual has higher fitness, the genetic information is spread to the population; otherwise it decreases in number and occasionally lost in the population. Dawkins discussed this process not from the viewpoint of individual which has function of self-reproduction but from the viewpoint of gene which is transferred generation to generation in the word “selfish gene” (Dawkins 1989). This view helps in understanding of the altruism and social behavior observed in living systems. That is, in such situation, some individuals may fail in reproduction of themselves, but genetic information can be well transferred to the next generation using population as a whole. He also generalized this concept to abstract self-reproductive processes and named information transferred by the self-reproductive systems “meme.” For example, some animals show cultural behaviors, and in such a situation, behavior found by an individual spreads to the population via learning by other individuals. Meme is a word to express such information to conduct a novel behavior which is reproduced in the population.

As we see the above, complexity in living systems and humankind emerged through adaptation processes. Requirements of adaptive systems can be summarized as the following three subprocesses:
  • Trial: The system should allow various trails seeking for a new configuration.

  • Evaluation: The performances of the trials should be evaluated.

  • Accumulation: Information of successful trials should be accumulated as the configuration of the system.

In genetic systems, the living system tries new configuration by genetic variation brought about by mutation and recombination of genetic information which brings about variation of phenotypes. Then it is evaluated by self-reproduction of individuals. Through the self-reproduction process, genetic information relating to the successful trial is accumulated in the set of genetic information.

As for cultural behavior observed in some animals, some new behavior is tried occasionally by some individual, and if it is beneficial, the individual learns it. Then, surrounding individuals imitate his/her behavior. Thus each individual evaluates the new behavior, and information is accumulated in individual nervous system through learning. In the economic system, as an example of social systems, some person or company tries new economic activities under their economic freedom. Its business is evaluated by the market, and its benefit enables expansion of its business. Information of new business is described and accumulated in many ways both inside the business and public.

Inference Mechanisms for Adaptation

The above adaptation process should also be discussed from a viewpoint of inference mechanisms that support the process. Basic inference in the evaluation process is induction. That is, performance of trial is evaluated and selected based on their actual success. However, even it started from natural selection, other mechanisms have been also employed. For example, sexual selection can be considered internalization of evaluation within species. It may accelerate evolution through enhancing selection pressure. However, it remains a question whether sexual selection is well consistent with natural selection. The nervous system and brain have enabled fast information processing and adaptation. With them, human being uses not only inductive inference but also deductive inference to predict results of trials through simulation before taking them actually.

As for making trials, it can be considered as abduction, that is, an inference of making hypotheses as promising trials. In genetic algorithms, alternatives are made mutation and crossover operations. Study of genetic algorithms showed importance of crossover operation. That is, making alternatives by combining building blocks acquired by evolution process is a key factor of successful application of GA. It highly depends on genetic representation of solutions and design of crossover operators.


This chapter gives a brief overview of complex systems. As guiding principles of complex behavior of physical systems, symmetry breaking in equilibrium systems and emergence of order in non-equilibrium systems are introduced as well as relating mathematical issues.

However, for understanding of complexities in biological and artificial systems requires, we discuss self-reproduction and adaptation as another guideline. That is, once a system acquires self-reproductive abilities, i.e., ability to make copies by themselves, adaptation in the sense of performance of self-reproduction starts. Self-reproduction and adaptation through it is a well-organized system in processing of information, i.e., trying alternatives, obtaining evaluation of them, and memorizing successful alternatives. The natural selection theory of evolution and artificial adaptive systems imitating it such as genetic algorithms are typical examples. Then various adaptive mechanisms would emerged in living systems on the base of self-reproductive system.

Further, human being constructed various complex systems such as sciences and technologies, including their application to engineered systems, and various social systems through their symbolic information processing by themselves using languages. It is also externalized with written languages, printing, telecommunication, computing, and automatic control technologies. Nowadays, with the progress of information and communication technologies supported by semiconductor, information processing ability of humankind is enhanced by further externalization of information processing. Automatic information processing with computers may lead to further evolution of complex systems.

Through this overview of the complex systems, we also notice that computational approaches of simulating such systems have played important role in study of the complex systems different from classical scientific methodologies such as mathematical analysis of, e.g., differential equations. It is quite plausible because complex systems emerged through adaptation and accumulation of information. However, computational approach to study complex systems has not established yet, and further development is needed in study of complex systems.


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Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Institute for Liberal Arts and SciencesKyoto UniversityKyotoJapan

Section editors and affiliations

  • Kyoichi Kijima
    • 1
  1. 1.Grad Sch of Decision Sci & TechTokyo Institute of Technology Grad Sch of Decision Sci & TechMeguro-kuJapan

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