  We study the small-world networks recently introduced by Watts and Strogatz
[Nature {\bf 393}, 440 (1998)], using analytical as well as numerical tools. We
characterize the geometrical properties resulting from the coexistence of a
local structure and random long-range connections, and we examine their
evolution with size and disorder strength. We show that any finite value of the
disorder is able to trigger a ``small-world'' behaviour as soon as the initial
lattice is big enough, and study the crossover between a regular lattice and a
``small-world'' one. These results are corroborated by the investigation of an
Ising model defined on the network, showing for every finite disorder fraction
a crossover from a high-temperature region dominated by the underlying
one-dimensional structure to a mean-field like low-temperature region. In
particular there exists a finite-temperature ferromagnetic phase transition as
soon as the disorder strength is finite.

Disordered Systems and Neural Networks