Crafting networks to achieve, or not achieve, chaotic states

TL;DR

This paper investigates how network topology influences the collective dynamics of the XY model, revealing that network dimension determines the presence of phase transitions and chaotic states.

Contribution

It introduces a network construction method controlling key topological parameters and analyzes their impact on the XY model's thermodynamic behavior, highlighting the role of network dimension.

Findings

Networks with dimension d<2 show no phase transitions.

Networks with dimension d>2 exhibit a second order phase transition.

Networks with dimension d=2 display chaotic and turbulent states.

Abstract

The influence of networks topology on collective properties of dynamical systems defined upon it is studied in the thermodynamic limit. A network model construction scheme is proposed where the number of links, the average eccentricity and the clustering coefficient are controlled. This is done by rewiring links of a regular one dimensional chain according to a probability $p$ within a specific range $r$, that can depend on the number of vertices $N$. We compute the thermodynamic behavior of a system defined on the network, the $XY-$rotors model, and monitor how it is affected by the topological changes. We identify the network dimension $d$ as a crucial parameter: topologies with $d\textless{}2$ exhibit no phase transitions while ones with $d\textgreater{}2$ display a second order phase transition. Topologies with $d=2$ exhibit states characterized by infinite susceptibility and macroscopic chaotic/turbulent dynamical behavior. These features are also captured by $d$ in the finite size context.