 Network theory is a way of describing the world in terms of a model called a network that allows us to capture the information about the relationships between things. But let's first think about why we might be interested in this at all. We often describe the world in terms of objects or things and their properties. We talk about a country and its GDP, people and their age, or the colour of a car. This type of component-based analysis works well when the system we're interested in is relatively isolated. But when we turn up the interactions and connectivity between elements within a system, it is increasingly the connections that come to shape the elements and define the system as a whole. And thus we need a model that captures this information about the relationships and allows us to reason about it. This is where network theory comes in. Network theory starts with a very simple view of the world as made up in nodes which are things or objects like people, cities, computers, etc. and the relationship between these things called edges such as friendships, trading partners, cables and so on. This abstract representation of the world can be used to model a wide variety of things, thus we can have social networks, biological networks or logistics networks composed of interacting suppliers and consumers. Network theory gives us a set of tools for analysing the individual elements and relations within these networks, the structure of the network and the properties that these different network structures give rise to. The first set of questions we might like to ask about a particular network relates to its degree of connectivity, that is how connected an individual element or the whole network is. This will tell us many things about it, such as how quickly a new event could spread or propagate through the system. The average degree of connectivity will give us a quick answer to this. This is calculated by taking the total number of edges and dividing it by the total number of nodes within the network. We also need to take into account how large the network is, that is to say how far it is on average from one point to another. This is called the average path length and we can calculate it by taking the average of all the path lengths between all the nodes. Because networks are all about connectivity, we often ascribe value to individual nodes based on their degree of connectivity. There are various methods for calculating this, but a popular one called eigenvector centrality measures both how many edges a node has and how connected the nodes it connects to are. Popular web search engines use variants of the eigenvector centrality measure to rank web pages by calculating both the number of links into a web page and the total degree of connectivity of the pages that link into them, thus gaining an idea of the relative importance of a website. Next, we're interested in talking about the overall structure of a network. This will be largely determined by how the relations between the nodes was formed. If the relation between the elements was generated randomly, we would expect a relatively even distribution of edges across the network. This type of structure or topology is called a distributed network, as the relative importance of any node is distributed across the entire network. A second type of network structure we can get is called decentralized or small world. This is generated by having local clusters of connections, but also having some random distant connections. An example of this might be a group of friends, with some of the friends having distant relatives in other parts of the world. By using these local connections within the group and distant connections, research has shown that it's possible to connect two random people within an average of just six steps and thus it is termed small world. Lastly, we have more centralized networks, called scale free networks. This is where many nodes have chosen to connect to the same node, giving a degree of connectivity that greatly exceeds the average, whilst leaving many with a very low level of connectivity. Many real world networks are thought to be scale free, including social, biological and technological systems such as the World Wide Web, where very few websites like Wikipedia have a very large amount of links into them, whilst the vast majority of websites have very few. These various types of network structures give rise to different properties. A key question we're interested in asking here is how robust or fragile is a particular type of network, as this will not only help us understand networks better, but it will also be of great significance in how we design and manage them. For example, think about a country with many small to medium sized cities supplying the population with various public services. If we were to remove one of the cities, it would have a limited effect on the overall system, because the network has a distributed structure, making it robust to failure of this kind. In contrary, if we take a country with one dominant capital city, with the rest of the urban network dependent upon it for core services, this centralized network may be more efficient, but it is also in what is called a more critical stage, as affecting this single primary node would have a large systemic effect. As we transit from an industrial to information societies, networks are merging as a new paradigm in how we structure our systems of organization, both social and technological. Network theory is a young and rapidly growing area that provides us with a set of tools for designing and managing these new types of organization, and more generally understanding the world around us from a different perspective.