In this seminar, I shall introduce a (relatively) novel way to characterize the macroscopic states of a dynamic model - the XY spin model - on networks. The method is based on the spectral decomposition of time series by using topological information about the underlying networks.
In the paper , we used three different classes of networks to generate time series 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. These signatures of the macroscopic states are independent of the underlying networks and can thus be used as universal signatures for the macroscopic states.
 S.d.N, P. Expert, T. Takaguchi and R. Lambiotte, Physical Review E 96, no. 1 (2017): 012312.