  I start by reviewing some basic properties of random graphs. I then consider
the role of random walks in complex networks and show how they may be used to
explain why so many long tailed distributions are found in real data sets. The
key idea is that in many cases the process involves copying of properties of
near neighbours in the network and this is a type of short random walk which in
turn produce a natural preferential attachment mechanism. Applying this to
networks of fixed size I show that copying and innovation are processes with
special mathematical properties which include the ability to solve a simple
model exactly for any parameter values and at any time. I finish by looking at
variations of this basic model.
