  Small world models are networks consisting of many local links and fewer long
range `shortcuts'. In this paper, we consider some particular instances, and
rigorously investigate the distribution of their inter--point network
distances. Our results are framed in terms of approximations, whose accuracy
increases with the size of the network. We also give some insight into how the
reduction in typical inter--point distances occasioned by the presence of
shortcuts is related to the dimension of the underlying space.
