A mathematical graph, or graph for short, is a structure representing pairwise relationships between objects. A graph is made up of nodes (sometimes called vertices, or points) which are connected by edges (sometimes called arcs, or lines). For example, nodes can embody people, maschines or individual steps in a process. Edges can illustrate the relations between nodes. They can can describe blood relations between people, data exchange between machines or give the interdependence of various steps in a chain of processes. The combination of nodes and the edges between them forms the graph.
In practical applications the term network is often used instead of graphs.
Once an application or a problem is transformed to the graph model, powerful concepts, algorithms and analysis techniques from Graph Theory and Network Science immediately apply. For this reason and their wide applicability, graphs and technologies based on graphs have become key elements of many important applications today. Some well-known applications include advertising in social networks, transportation networks or data management applications.
The following example shows how interpreting data as a graph can help us make better sense of structured data. Clearly, the better we understand the complex data we work with the easier it will be to build applications on top of it. In this case, we look at a simple communication network, i.e. people sending emails to each other. However, it is important to emphasize that virtually everything can be seen as a graph.
Finding the best way to model data as a graph and make maximum use of it in an application is a challenging yet important problem.