In this module, we will be trying to define what exactly a complex system is. Before we start, we should note that there is no formal definition for what a complex system is, and thus there remains many different perspectives and opinions on the subject. What we present here is just a working definition. Firstly, a complex system is a special class of system. A system is simply a set of parts called elements, and a set of connections between these parts called relations. These parts can be ordered or unordered. An unordered system is simply a set of things. Because there is no specific structure or order, we can describe a set simply by listing all of its elements and their properties. So a pile of stones on the ground is an example of an unordered set. As there is no pattern or order to the system, we can only describe it by describing the properties of each element in isolation and then adding them all up, with the whole set being nothing more than the sum of its individual parts. If, in contrast, through the relations, these parts are ordered in a specific way, then they can function together as an entirety, and out of these parts working together, we get the emergence of a global pattern of organization that is capable of functioning as a coherent whole. For example, if all the parts in our car are arranged in a specific way, then we will have the global functionality of a vehicle of transportation, or out of the specific arrangement of billions of cells, and the different specialized organs that make up our body, we get the emergence of a global system that enables us to operate as an entire organism. So that is the basic model of a system. It consists of elements and relations. When those elements work together, we get the emergence of a new level of organization. Now, let's start adding complexity to this. Probably the only property that will be in all definitions of a complex system is that they consist of many parts. Many parts that are distributed out without centralized control. Organization is formed out of the local interactions between the parts through a process of self-organization that gives rise to the emergence of new levels of organization. With the phenomenon of emergence that we were previously discussing, a whole new level to the system has developed, which then starts to interact with other systems in its environment, the result being that new patterns of organization develop, and once again we get the emergence of another level of organization, and so on. People form part of social groups that form part of broader society, which in turn forms part of humanity. The point to take away here is that these systems have a hierarchical structure. This is a pervasive phenomenon in our world. Elements are nested inside of subsystems, which in turn form part of larger systems, and so on. All complex systems have this multi-dimensional property to them. They are composed of many elements on many different scales, with all of these levels affecting each other. A business is part of a local economy, which is part of a national economy, which in turn is part of a global economy. Each is interconnected and interdependent with the others. We cannot fully isolate one component or reduce the whole thing to one level, and this is a primary source of complexity. So this is our first property to a complex system, many different parts that are distributed out, with local interactions and self-organization giving rise to new emergent levels on different scales. Next, interdependence and non-linearity. The parts to a complex system are highly interdependent, and this interdependence creates non-linearity. Almost all well-formulated definitions for complex systems involve the term non-linearity. It is a continuously recurring and pervasive theme. Non-linearity arises from the fact that when we put two or more things together, the result may not necessarily be a simple addition of each element's properties in isolation. In contrary, we may get a combined effect that is greater or less than the simple sum of each part because of their interdependent nature. Examples of this might be two sound waves that are perfectly out of sync, canceling each other out through noise interference, or the division of labor as can be seen in many human and insect communities, resulting in synergies, which means the output will be far greater than what individuals could accomplish in isolation. Due to what are called feedback loops, non-linear systems may grow or decay at an exponential rate. These periods of rapid change are defined as phase transitions. Thus, complex systems are known to be able to shift or flip into whole new regimes within very brief periods of time. Some small change in input value to the system can, through feedback loops, trigger a large systemic effect. This is called sensitivity to initial conditions, and it is the central idea within chaos theory. Examples of this butterfly effect can be seen in financial crises and the collapse of ecosystems such as coral reefs. Next, connectivity. Many definitions for complex systems involve dense or high levels of interconnectivity between components. As we turn up the degree of connectivity, it becomes the nature and structure of these connections that define the system as opposed to the properties of its components. How are things connected and what is connected to what become the main questions? At some critical level of connectivity, the system stops being a set of parts and becomes a network of connections, and it is now all about how things flow in this network. Networks are the true geometry of complex systems, and this connectivity reshapes our traditional three-dimensional Euclidean conception of space. In complex systems like the global air transportation system, or the financial system, or the internet, space is redefined in terms of topology created by connectivity. What matters is your position in the network structure and your degree of connectivity. Connectivity again leads us into the world of complexity as the number of relations between elements can grow in an exponential fashion. If we take just a handful of elements, they can become connected in possibly thousands or even millions of different ways. Lastly, autonomy and adaptation. Whether we are talking about a flock of birds, the internet, or our global economy, there is no top-down centralized mechanism for coordinating the whole system. Within complex systems, elements have a degree of autonomy, often through their capacity to adapt to their local environment according to their own set of instructions. Without centralized coordination and with a degree of autonomy, elements can synchronize their states locally or cooperate, resulting in the emergence of patterns of organization from the bottom up. With autonomy and adaptation also comes the capacity for a variety of different responses for any given phenomenon, meaning complex systems are often heterogeneous with high levels of diversity. Ecosystems and multicultural societies are good examples of this. Without centralized coordination, complex systems develop on the macro scale through a process of evolution. Elements within complex adaptive systems are subject to the evolutionary force of selection, where those that are best suited to that environment are selected and replicated while others are not. Products are subjected to selection within a market environment. In democracies, politicians are subject to selection by voters, and creatures in ecosystems are subjected to natural selection through competition. In such a way, the whole macro scale system manages to adapt to its environment without centralized coordination and develop to exhibit higher levels of both differentiation and integration. The greater the autonomy and capacity for adaptation that elements have, the more complex the system we are dealing with. In summary, we have been trying to lay down a basic working definition for a complex system while remembering that there is no formal consensus on the subject. We firstly talked about how a complex system is a special class of system. We defined a system as a set of elements and the relations between them. We saw how when these parts are arranged in a specific order for them to function as an entirety, we get what is called the process of emergence, whereby a new level of organization emerges. We then began to add complexity to our model of a system by defining it as a product of four primary parameters. Firstly, talking about the number of elements and different levels to the hierarchy within our system. We then discussed non-linearity as another dimension to complexity, where non-additive interactions and feedback loops over time can give us exponential relations between the input and output to a system and lead to phase transitions. We also talked about connectivity as another driver of complexity, as heightened connectivity within complex systems means they often appear to us as networks. Lastly, we discussed how autonomy and adaptation enable self-organization in the process of evolution that shapes complex systems on the macro scale.