Graph spectral characterisation of the XY model on complex networks

TL;DR

This paper introduces a spectral method based on the Laplacian eigenbasis to characterize the macroscopic states of the XY spin model on complex networks, providing universal signatures independent of network topology.

Contribution

It presents a novel spectral decomposition approach using temporal Graph Signal Transform to identify macroscopic states of the XY model on various complex networks.

Findings

Spectral signatures are consistent across different network types.

The method effectively distinguishes between the three macroscopic states.

Universal patterns of activity are identified through spatial power spectra.

Abstract

There is recent evidence that the $XY$ spin model on complex networks can display three different macroscopic states in response to the topology of the network underpinning the interactions of the spins. In this work, we present a novel way to characterise the macroscopic states of the $XY$ spin model based on the spectral decomposition of time series using topological information about the underlying networks. We use three different classes of networks to generate time series of the spins for the three possible macroscopic states. We then use the temporal Graph Signal Transform technique to decompose the time series of the spins on the eigenbasis of the Laplacian. From this decomposition, we produce spatial power spectra, which summarise the activation of structural modes by the non-linear dynamics, and thus coherent patterns of activity of the spins. These signatures of the macroscopic states are independent of the underlying networks and can thus be used as universal signatures for the macroscopic states. This work opens new avenues to analyse and characterise dynamics on complex networks using temporal Graph Signal Analysis.