1999/03/31 by A. Barrat, M. Weigt · 2 citations
Physics and Astronomy · #cond-mat.dis-nn
published as Europ. Phys. J. B 13, 547 (2000) · 19 pages including 15 figures, version accepted for publication in EPJ B
arxiv created 1999/08/25 · arxiv updated 2009/11/30
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.