On the properties of small-world network models
Abstract
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.
Cite
@article{arxiv.cond-mat/9903411,
title = {On the properties of small-world network models},
author = {A. Barrat and M. Weigt},
journal= {arXiv preprint arXiv:cond-mat/9903411},
year = {2007}
}
Comments
19 pages including 15 figures, version accepted for publication in EPJ B