Exact results and new insights for models defined over small world networks. First and second order phase transitions. II: Applications

TL;DR

This paper applies a novel analytical method to small-world network models, revealing detailed phase transition behaviors including second-order and first-order transitions in specific models.

Contribution

It extends the novel method from part I to analyze multiple small-world models, uncovering new insights into their phase transition properties.

Findings

Identification of second-order spin glass phase transition in the 1D small-world model

Discovery of two critical temperatures indicating first- and second-order phase transitions

Analytical solutions for small-world models in various dimensions

Abstract

We apply a novel method (presented in part I) to solve several small-world models for which the method can be applied analytically: the Viana-Bray model (which can be seen as a 0 or infinite dimensional small-world model), the one-dimensional chain small-world model, and the small-world spherical model in generic dimension. In particular, we analyze in detail the one-dimensional chain small-world model with negative short-range coupling showing that in this case, besides a second-order spin glass phase transition, there are two critical temperatures corresponding to first- or second-order phase transitions.