As Sylvester observed in the 19th century, graphs are a useful tool for modelling the connectivity structure of various types of networks in a way that allows researchers to focus solely on the connectivity properties of the network. Sylvester studied networks for which the number of nodes was in the single digits. The graphs which arise in the study of 21st century networks are much larger. For example, the graph formed by the hyperlinks between webpages has billions of nodes.
We present some tools which allow one to determine the global structure of a large network by examining it locally, and some results obtained by applying them. The talk assumes no prior knowledge of graph theory.