 When we hear the word network, all sorts of things spring to mind, like social networks and the internet in particular. But the power of network theory is really in its high degree of abstraction. So the first thing for us to do is to try and start back at the beginning by forgetting about what we think we know about networks, and embracing the abstract language of networks, what we call graph theory. In the formal language of mathematics, a network is called a graph, and graph theory is the area of mathematics that studies these objects called graphs. The first theory of graphs goes back to 1736. The first textbook came about in 1958, but most of the work within this field is less than a few decades old. In its essence, a graph is really a very simple thing. It consists of just two parts, what are called vertices and edges. Firstly, vertices. A vertex, otherwise called a node, is a thing. That is to say, it is an entity and we can ascribe some value to it. So a person is an example of a node, as is a car, planet, farm, city, or molecule. All of these things have static properties that we can quantify, such as the color of our car, the size of our farm, or the weight of our molecule. Within network science, vertices are more often called nodes, so we'll be typically using this term during the course. Edges can be defined as a relation of some sort between two nodes. This connection may be tangible as in cables between computers on a network, or the roads between cities within a national transportation system, or edges may be intangible, such as social relations of friendship. Edges may also be called links, ties, or relations, and we will be more often using this latter term during the course. The nodes belonging to an edge are called the ends, endpoints, or end vertices of the edge. Within graph theory, networks are called graphs, and a graph is defined as a set of edges and a set of vertices. A simple graph does not contain loops or multiple edges, but a multigraph is a graph with multiple edges between the nodes. So whereas a simple graph of a transportation system would just tell us whether there is a connection between two cities, a multigraph would instead show us all the different connections between the two cities. A graph can be directed or undirected. With an undirected graph, edges have no orientation. For example, a diplomatic relation between two nations may be mutual and thus have no direction to the edge between the nodes. These undirected graphs have unordered pairs of nodes. That means we can just switch them around. If Jane and Paul are married, we can say Jane is married to Paul, or we can say Paul is married to Jane. It makes no difference, and thus it is an unordered pair. In contrast to an undirected graph, we have directed graphs, which is the set of nodes connected by edges where the edges have a direction associated with them. This is typically denoted with arrows indicating the direction. For example, if we were drawing a graph of international trade, the graph might have arrows to indicate the direction of the flow of goods and services. So directed graphs have some order to the relations between the nodes, and this can be quite important. A graph is a weighted graph if we associate a number to each edge. These numbers quantify the degree of interaction between the nodes or the volume of exchange. So with our trading example earlier, if we wanted to convert this into a weighted graph, we would then ascribe a quantitative value to the amount of trade between the different nations. So this is the basic language of graphs, but we can extend this language to talk about graphs that have multiple types of nodes and edges. What are called multiplex networks that will add a whole new level of complexity to our representation, allowing us to capture how different networks interrelate and overlap to affect each other. But this is beyond the scope of our course, as the basic language we have outlined above will be sufficed for our introduction to network theory.