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Mathematics LibreTexts

1.2: Topical Clusters

 

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Nonlinear dynamics is probably the topical cluster that has the longest history, at least from as far back as the 17th century when Isaac Newton and Gottfried Wilhelm Leibniz invented calculus and differential equations. But it was found only in the 20th century that systems that include nonlinearity in their dynamics could show some weird behaviors, such as chaos [5, 6] (which will be discussed later). Here, nonlinearity means that the outputs of a system are not given by a linear combination of the inputs. In the context of system behavior, the inputs and outputs can be the current and next states of the system, and if their relationship is not linear, the system is called a nonlinear system. The possibility of chaotic behavior in such nonlinear systems implies that there will be no analytical solutions generally available for them. This constitutes one of the several origins of the idea of complexity.

Systems theory is another important root of complex systems science. It rapidly developed during and after World War II, when there was a huge demand for mathematical theories to formulate systems that could perform computation, control, and/or communication. This category includes several ground-breaking accomplishments in the last century, such as Alan Turing’s foundational work on theoretical computer science [7], Norbert Wiener’s cybernetics [8], and Claude Shannon’s information and communication theories [9]. A common feature shared by those theories is that they all originated from some engineering discipline, where engineers were facing real-world complex problems and had to come up with tools to meet societal demands. Many innovative ideas of systems thinking were invented in this field, which still form the key components of today’s complex systems science.

Game theory also has an interesting societal background. It is a mathematical theory, established by John von Neumann and Oskar Morgenstern [10], which formulates the decisions and behaviors of people playing games with each other. It was developed during the Cold War, when there was a need to seek a balance between the two mega powers that dominated the world at that time. The rationality of the game players was typically assumed in many game theory models, which made it possible to formulate the decision making process as a kind of deterministic dynamical system (in which either decisions themselves or their probabilities could be modeled deterministically). In this sense, game theory is linked to nonlinear dynamics. One of the many contributions game theory has made to science in general is that it demonstrated ways to model and analyze human behavior with great rigor, which has made huge influences on economics, political science, psychology, and other areas of social sciences, as well as contributing to ecology and evolutionary biology.

Later in the 20th century, it became clearly recognized that various innovative ideas and tools arising in those research areas were all developed to understand the behavior of systems made of multiple interactive components whose macroscopic behaviors were often hard to predict from the microscopic rules or laws that govern their dynamics. In the 1980s, those systems began to be the subject of widespread interdisciplinary discussions under the unified moniker of “complex systems.” The research area of complex systems science is therefore inherently interdisciplinary, which has remained unchanged since the inception of the field. The recent developments of complex systems research may be roughly categorized into four topical clusters: pattern formation, evolution and adaptation, networks, and collective behavior.

Pattern formation is a self-organizing process that involves space as well as time. A system is made of a large number of components that are distributed over a spatial domain, and their interactions (typically local ones) create an interesting spatial pattern over time. Cellular automata, developed by John von Neumann and Stanisław Ulam in the 1940s [11], are a well-known example of mathematical models that address pattern formation. Another modeling framework is partial differential equations (PDEs) that describe spatial changes of functions in addition to their temporal changes. We will discuss these modeling frameworks later in this textbook.

Evolution and adaptation have been discussed in several different contexts. One context is obviously evolutionary biology, which can be traced back to Charles Darwin’s evolutionary theory. But another, which is often discussed more closely to complex systems, is developed in the “complex adaptive systems” context, which involves evolutionary computationartificial neural networks, and other frameworks of man-made adaptive systems that are inspired by biological and neurological processes. Called soft computing, machine learning, or computational intelligence, nowadays, these frameworks began their rapid development in the 1980s, at the same time when complex systems science was about to arise, and thus they were strongly coupled—conceptually as well as in the literature. In complex systems science, evolution and adaptation are often considered to be general mechanisms that can not only explain biological processes, but also create non-biological processes that have dynamic learning and creative abilities. This goes well beyond what a typical biological study covers.

Finally, networks and collective behavior are probably the most current research fronts of complex systems science (as of 2015). Each has a relatively long history of its own. In particular, the study of networks was long known as graph theory in mathematics, which was started by Leonhard Euler back in the 18th century. In the meantime,the recent boom of network and collective behavior research has been largely driven by the availability of increasingly large amounts of data. This is obviously caused by the explosion of the Internet and the WWW, and especially the rise of mobile phones and social media over the last decade. With these information technology infrastructures, researchers are now able to obtain high-resolution, high-throughput data about how people are connected to each other, how they are communicating with each other, how they are moving geographically, what they are interested in, what they buy, how they form opinions or preferences, how they respond to disastrous events, and the list goes on and on. This allows scientists to analyze the structure of networks at multiple scales and also to develop dynamical models of how the collectives behave. Similar data-driven movements are also seen in biology and medicine (e.g., behavioral ecology, systems biology, epidemiology), neuroscience (e.g., the Human Connectome Project [12]), and other areas. It is expected that these topical areas will expand further in the coming decades as the understanding of the collective dynamics of complex systems will increase their relevance in our everyday lives.

Here, I should note that these seven topical clusters are based on my own view of the field, and they are by no means well defined or well accepted by the community. There must be many other ways to categorize diverse complex systems related topics. These clusters are more or less categorized based on research communities and subject areas, while the methodologies of modeling and analysis traverse across many of those clusters. Therefore, the following chapters of this textbook are organized based on the methodologies of modeling and analysis, and they are not based on specific subjects to be modeled or analyzed. In this way, I hope you will be able to learn the “how-to” skills systematically in the most generalizable way, so that you can apply them to various subjects of your own interest.

Exercise \(\PageIndex{1}\)

Choose a few concepts of your own interest from Fig. 1.1.1. Do a quick online literature search for those words, using Google Scholar (http://scholar.google.com/), arXiv (http://arxiv.org/), etc., to find out more about their meaning, when and how frequently they are used in the literature,  and in what context.

Exercise \(\PageIndex{2}\)

Conduct an online search to find visual examples or illustrations of some of the concepts shown in Fig. 1.1.1. Discuss which example(s) and/or illustration(s) are most effective in conveying the key idea of each concept. Then create a short presentation of complex systems science using the visual materials you selected.

Exercise \(\PageIndex{3}\)

Think of other ways to organize the concepts shown in Fig.1.1.1 (and any other relevant concepts you want to include). Then create your own version of a map of complex systems science.

Now we are ready to move on. Let’s begin our journey of complex systems modeling and analysis.