  Small-world networks---complex networks characterized by a combination of
high clustering and short path lengths---are widely studied using the
paradigmatic model of Watts and Strogatz (WS). Although the WS model is already
quite minimal and intuitive, we describe an alternative formulation of the WS
model in terms of a distance-dependent probability of connection that further
simplifies, both practically and theoretically, the generation of directed and
undirected WS-type small-world networks. In addition to highlighting an
essential feature of the WS model that has previously been overlooked, this
alternative formulation makes it possible to derive exact expressions for
quantities such as the degree and motif distributions and global clustering
coefficient for both directed and undirected networks in terms of model
parameters.
