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 seminar, I
shall introduce a different 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.
In this work [1], 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 summarise
the activation of structural modes by the non-linear dynamics, and thus
coherent patterns of activity of the spins. Remarkably, these signatures of
the macroscopic states are independent of the underlying networks and can
thus be used as universal signatures for the macroscopic states.

[1] S.d.N, P. Expert, T. Takaguchi and R. Lambiotte, Phys. Rev. E,
96(1):012312, 2017