Small-world networks: Evidence for a crossover picture

TL;DR

This paper investigates the transition to small-world behavior in networks, proposing it as a crossover phenomenon influenced by network size and disorder level, with a specific scaling law for the crossover size.

Contribution

It introduces a scaling framework showing small-world properties emerge as a crossover, not a phase transition, depending on network size and disorder.

Findings

Crossover size scales as n* ∼ p^(-2/3) for small p

Small-world behavior is a crossover, not a phase transition

Average distance scales with n / n*

Abstract

Watts and Strogatz [Nature 393, 440 (1998)] have recently introduced a model for disordered networks and reported that, even for very small values of the disorder $p$ in the links, the network behaves as a small-world. Here, we test the hypothesis that the appearance of small-world behavior is not a phase-transition but a crossover phenomenon which depends both on the network size $n$ and on the degree of disorder $p$. We propose that the average distance $\ell$ between any two vertices of the network is a scaling function of $n / n^*$. The crossover size $n^*$ above which the network behaves as a small-world is shown to scale as $n^*(p \ll 1) \sim p^{-\tau}$ with $\tau \approx 2/3$.