Predicting synchronization regimes with dimension reduction on modular graphs

We study the synchronization of oscillator networks with community structures, such as two-star graphs and networks generated by the stochastic block model (SBM). We map high-dimensional (complete) dynamics unto low-dimensional (reduced) dynamics that captures the mesoscale features. This is achieved by introducing a spectral method for dimension reduction that generalizes and improves the method recently proposed in [1]. For the Kuramoto model on realizations of the SBM, the reduced dynamics describes the synchronized states correctly and, as a bonus, reveals the detectability limit for the SBM [2] [see FIG. (a)]. For the SakaguchiKuramoto (SK) model on the mean adjacency matrix of the SBM, we prove that the spectral method leads to the same reduced model as the one obtained with the Ott-Antonsen Ansatz [3], highlighting the consistency of the approach with previous works. Moreover, we find new regions in the structural parameter space admitting chimeras, dynamical states characterized by the cohabitation of full synchronization in one community and partial synchronization in others. For the SBM, the size of chimera regions in the parameter space reaches a maximum when the communities are slightly asymmetric in size [FIG. (b) (Top)]. However, for the two-star graphs, the size of the chimera regions has a more complicated behavior and exhibits multiple step transitions with respect to the ratio of the periphery sizes [FIG. (b) (Bottom)].