On the properties of small-world network models

A. Barrat; M. Weigt

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arXiv · cond-mat/9903411

On the properties of small-world network models

A. Barrat·M. Weigt

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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.

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A. Barrat, M. Weigt. 1999-08-25. On the properties of small-world network models. https://arxiv.org/abs/cond-mat/9903411

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