  In this paper we study the small-world network model of Watts and Strogatz,
which mimics some aspects of the structure of networks of social interactions.
We argue that there is one non-trivial length-scale in the model, analogous to
the correlation length in other systems, which is well-defined in the limit of
infinite system size and which diverges continuously as the randomness in the
network tends to zero, giving a normal critical point in this limit. This
length-scale governs the cross-over from large- to small-world behavior in the
model, as well as the number of vertices in a neighborhood of given radius on
the network. We derive the value of the single critical exponent controlling
behavior in the critical region and the finite size scaling form for the
average vertex-vertex distance on the network, and, using series expansion and
Pade approximants, find an approximate analytic form for the scaling function.
We calculate the effective dimension of small-world graphs and show that this
dimension varies as a function of the length-scale on which it is measured, in
a manner reminiscent of multifractals. We also study the problem of site
percolation on small-world networks as a simple model of disease propagation,
and derive an approximate expression for the percolation probability at which a
giant component of connected vertices first forms (in epidemiological terms,
the point at which an epidemic occurs). The typical cluster radius satisfies
the expected finite size scaling form with a cluster size exponent close to
that for a random graph. All our analytic results are confirmed by extensive
numerical simulations of the model.
