 In this video we will be giving an overview to the area of complexity theory by looking at the major theoretical frameworks that are considered to form part of it and contribute to the study of complex systems. Complexity theory is a set of theoretical frameworks used for modeling and analyzing complex systems within a variety of domains. Complexity has proven to be a fundamental feature of our universe that is not amenable to our traditional methods of modern science. And thus as researchers have encountered it within many different areas from computer science, to ecology, to engineering, they have had to develop new sets of models and methods for approaching it. Out of these different frameworks has emerged a core set of commonalities that over the past few decades has come to be recognized as a generic framework for studying complex systems in the abstract. Complexity theory encompasses a very broad and very diverse set of models and methods. As yet there is no proper formulation to structure and give definition to this framework. Thus we will present it as a composite of our four main areas that encompass the different major perspectives on complex systems and how to best interpret them. Including self-organization theory, non-linear systems, network theory, and adaptive systems theory. Firstly, self-organization and emergence. Complex systems are composed of many small parts without centralized control, examples being flocks of birds, financial markets, social networks, global logistics networks, or the human brain. Without centralized control, global organization is an emergent feature of the local interactions between the parts. Whereas the term emergence is a general concept referring to how new levels of organization are formed as we put component parts together, the theory of self-organization presents a number of concrete models for understanding how this process takes place. The model of self-organization draws upon information theory to understand organization in terms of information and entropy. It draws upon ideas in physics surrounding synchronization and pattern formation and ideas in chemistry surrounding dissipative systems and far from equilibrium processes. Here we are looking at how elements governed by simple rules synchronize their behavior with the result being a process of self-organization as patterns of organization emerge from the bottom up. Researchers try to model complex systems by capturing these local rules and using computational tools like agent-based modeling to try and simulate the process through which order emerges out of initially homogeneous or disordered states. Next, non-linear systems and chaos theory. Non-linearity is an inherent feature and major theme that crosses all areas of complex systems. A lot of non-linear systems theory has its origins in quite dense and obscure mathematics and physics. Out of the study of certain types of equations, weather patterns, fluid dynamics, and particular chemical reactions has emerged some very counter-intuitive phenomena in the form of the butterfly effect and chaos. Chaos theory, which is the study of non-linear dynamical systems, was one of the first major challenges to the Newtonian paradigm that was accepted into the mainstream body of scientific knowledge. Our modern scientific framework is based upon linear systems theory and this places significant constraints upon it. Linear systems theory is dependent upon the concept of a system having an equilibrium. Although linear systems theory often works as an approximation, the fact is that many of the phenomena we are interested in describing are non-linear. Processes of change, such as regime shifts within ecosystems and society, happen far from equilibrium. They are governed by the dynamics of feedback loops and not linear equations. Trying to model complex systems by using traditional linear systems theory is like trying to put a screw into a piece of wood with a hammer. We are simply using the wrong tool because it is the only one we have. Thus, the areas of non-linear systems and their dynamics is another major part of the framework of complexity theory that has come largely from physics, mathematics, and the study of far from equilibrium processes in chemistry. Next, network theory. Network theory is another major area to complexity theory as almost all complex systems can be understood and modeled effectively as networks. Network theory is a formal mathematical language, but it has proven a very practical tool for analysis and thus has found widespread application in many areas. The study of networks is probably the youngest and most active area of complexity science, again driven by the rise of computation and the fundamental role that networks are starting to play in our world with the advent of information technology. With the theory of networks and the availability of new sources of data, we are starting to get a real picture to what some of these complex systems that make up our world actually look like. We can start to see the connections within financial systems through which contagion spreads, the real-time movement of freight around the globe, or sociopolitical networks that influence our lives. This is a new kind of science, driven less by models and equations, but more by real-time dense data sets. This means we are no longer left staring at models, but now have accessible visualizations to give us a much more rich, intuitive, and in many ways, real sense for what exactly these complex systems are like. The main contributions to this area have come from the area of mathematics called graph theory, and again, computer science. The last major area to complexity theory that we will discuss is that of complex adaptive systems. Complex adaptive systems are classical examples of complex systems, and people often use the two words somewhat interchangeably. They consist of many parts acting and reacting to each other's behavior, like a school of fish swimming together, nation states within the international political environment, or businesses in a market. They are highly dynamic and develop through an evolutionary-like process. The central issue is that of the process of adaptation and evolution. The idea of adaptation formed a central part of cybernetics that contributed ideas surrounding control systems and how systems regulate themselves and their environment in order to maintain homeostasis. A key issue here is that of the dynamics surrounding cooperation and competition that form as adaptive agents interact and try to pursue their goals collectively. One could also include game theory here, a branch of mathematics for modeling the interaction between adaptive agents of all kinds and the dynamics of cooperation and competition that form out of this. When the idea of adaptation is generalized to a whole population of agents and takes place over a series of life cycles, it can be termed evolution. And the theory of evolution is one of the major contributions that ecology has made to complexity theory. We now have a number of different models for understanding evolution including evolutionary game theory, replicator equations, fitness landscapes, and genetic algorithms, among others. This is an area that has grown out of cybernetics, computer science, economics, and ecology. Lastly, we'll discuss a little of the context and significance of the area of complexity theory as it plays a somewhat unique role within the framework of contemporary science. The website Scholarpedia describes complexity theory as an emerging post-Newtonian paradigm. There is a lot packed into this short statement, so let's try and unravel and make sense of it. The Newtonian framework is based on linear systems theory. This has been a powerful tool for helping us understand the world. Through the contributions of millions of researchers over the course of centuries, we have built up a large and sophisticated body of scientific knowledge, which is one of humanity's greatest achievements. Throughout the 20th century, though, the Newtonian paradigm and linear systems theory have become increasingly called into question as general relativity and then chaos theory proved some of its most basic assumptions to be, in fact, flawed. The fact is that much of the phenomena that we are really interested in are inherently nonlinear, such as almost all sociopolitical, ecological, and economic phenomena. A core challenge of 21st century science, then, is to extend this framework into the world of nonlinear systems and complexity, and this means going beyond the Newtonian framework. As Scholarpedia puts it, developing a post-Newtonian paradigm, and this is exactly what complexity theory is doing. To summarize, we have been giving a quick overview to the area of complexity theory, which we defined as a set of theoretical frameworks used for modeling complex systems within a variety of domains. We looked at four of the major modeling frameworks that fall under its canopy. We firstly talked about self-organization and how it gives us the tools to understand the process of emergence where global patterns form out of only local interactions. We talked about the theory of nonlinear systems and how it has emerged out of the study of chaotic physical and chemical processes. We then discussed network theory as another major domain that understands complex systems in terms of connectivity and how things flow through these systems. The last major area we looked at was the theory of adaptive systems that tries to understand complex adaptive systems in terms of the interaction between adaptive agents, cooperation and competition, and the dynamics of evolution. Finally, we tried to provide some insight into the significance and context of complexity theory as a so-called post-Newtonian paradigm as it tries to extend our scientific body of knowledge into the world of nonlinear systems.