Graph spectral characterization of the XY model on complex networks.

Paul Expert,Taro Takaguchi,Sarah De Nigris,Renaud Lambiotte

doi:10.1103/physreve.96.012312

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 way to characterize 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 summarize the activation of structural modes by the nonlinear dynamics, and thus coherent patterns of activity of the spins. These signatures of the macroscopic states are independent of the underlying network class and can thus be used as robust signatures for the macroscopic states. This work opens avenues to analyze and characterize dynamics on complex networks using temporal Graph Signal Analysis.

Highlights

Activity of brain regions [1], car flow on roads [2], meta-population epidemic [3], all these arguably very different systems have in common that they can be represented as the activity of a quantity of interest on the nodes of a network
We apply the temporal Graph Signal Transform (TGST) using the Laplacian eigenbasis to the time series generated by the XY spin model in the three stationary phases on the three network models
In this paper we presented the temporal Graph Signal Transform, a method to decompose time-dependent signals existing on the nodes of a network, using a basis that incorporates structural information

Summary

Introduction

Activity of brain regions [1], car flow on roads [2], meta-population epidemic [3], all these arguably very different systems have in common that they can be represented as the activity of a quantity of interest on the nodes of a network. The coupling between the dynamics on the nodes and their network of interactions often leads to emergent collective states In simple cases, such as the Kuramoto and XY spin models, these macroscopic states can be classified according to the behavior of an order parameter that measures the global coherence of the units comprising the system. This order parameter is blind to the structure of the underlying interaction network and does not allow one to investigate how the system behaves at different structural scales. Having such a method will help us to understand the functioning and mitigate disruption of complex systems or even engineer new ones

Methods

Results

Conclusion