On the properties of small-world network models

We study the small-world networks recently introduced by Watts and Strogatz [Nature 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 ev olution with size and disorder strength. We show that any finite value of t he disorder is able to trigger a "small-world" behaviour as soon as the ini tial lattice is big enough, and study the crossover between a regular latti ce and a "small-world" one. These results are corroborated by the investiga tion of an Ising model defined on the network, showing for every finite dis order fraction a crossover from a high-temperature region dominated by the underlying one-dimensional structure to a mean-field like low-temperature r egion. In particular there exists a finite-temperature ferromagnetic phase transition as soon as the disorder strength is finite.