On the properties of small-world network models

TL;DR

This paper analyzes the properties of small-world networks introduced by Watts and Strogatz, combining analytical and numerical methods to understand their geometrical features, evolution, and phase transition behavior.

Contribution

It provides a detailed characterization of small-world network properties and their evolution, including the impact of finite disorder on phase transitions, supported by analytical and numerical evidence.

Findings

Any finite disorder induces small-world behavior in large enough networks.

There is a crossover from regular lattice to small-world structure as disorder increases.

A finite-temperature ferromagnetic phase transition occurs at finite disorder.

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.