Graph spectral characterisation of the XY model on complex networks

There is recent evidence [1,2,3] 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 seminar, I shall introduce an alternative way to characterise those macroscopic states of the XY spin model based on the spectral decomposition of time series using topological information about the underlying networks.
In this work [4], we use three different classes of networks to generate time series of the spins for the three possible macroscopic states. We then apply 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 summarises 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 analysis opens new avenues to analyse and characterise dynamics on complex networks using temporal Graph Signal Analysis.

[1] Sarah de Nigris and Xavier Leoncini. Crafting networks to achieve or not achieve chaotic states. Phys. Rev. E, 91:042809, 2015.
[2] Sarah De Nigris and Xavier Leoncini. Emergence of a non-trivial fluctuating phase in the XY-rotors model on regular networks. EPL, 101(1):10002, 2013.
[3] Sarah De Nigris and Xavier Leoncini. Critical behavior of the XY-rotor model on regular and small-world networks. Phys. Rev. E, 88:012131, 2013.
[4] Sarah de Nigris, Paul Expert, Renaud Lambiotte, and Taro Takaguchi. Graph spectral characterisation of the XY model on complex networks. Phys. Rev. E, 96(1):012312, 2017.