First-order transition in small-world networks

TL;DR

This paper demonstrates that the small-world network transition is a first-order phase transition characterized by a discontinuity in the shortest-path distance, confirmed through numerical simulations across multiple dimensions.

Contribution

The study provides numerical evidence that the small-world transition is a first-order transition and establishes a relation between finite-size effects and the transition's nature.

Findings

Transition is first-order with a discontinuity in shortest-path distance.

Finite-size effects scale as L^{-d}, defining a persistence size L* ~ p^{-1/d}.

Numerical simulations in 1-4 dimensions confirm the theoretical scaling relation.

Abstract

The small-world transition is a first-order transition at zero density $p$ of shortcuts, whereby the normalized shortest-path distance undergoes a discontinuity in the thermodynamic limit. On finite systems the apparent transition is shifted by $\Delta p \sim L^{-d}$. Equivalently a ``persistence size'' $L^* \sim p^{-1/d}$ can be defined in connection with finite-size effects. Assuming $L^* \sim p^{-\tau}$, simple rescaling arguments imply that $\tau=1/d$. We confirm this result by extensive numerical simulation in one to four dimensions, and argue that $\tau=1/d$ implies that this transition is first-order.