The ASC Networks cluster visiting researcher Prof. Richard Wilson (Department of Computer Science, University of York) gives a public lecture on his work on networks at Keble College. A characterisation of a network is a number which describes some structural property of the network; a good example is the number of edges. In this talk I will discuss a wide range of characterisations of networks which we have developed at York over the past decade. The inspiration for these comes from diverse areas of mathematics and physics, and I will explore ideas from graph spectra, random walks (classical and quantum), heat diffusion and the connection between prime numbers and networks. These all lead to characterisations which reveal different properties of the networks.