Heterogeneous nucleation in a multiplex adaptive network.

We investigate the phase transition to synchronization in a multiplex network of adaptively coupled Kuramoto oscillators with frequency disorders. We demonstrate how the frequency disorder in a specific layer initiates nucleation across the layers of the multiplex networks. We find that the multiplex network mostly exhibits an abrupt single-step phase transition with occasional gradual multistep phase transition under uniform frequency distribution in all the layers. However, the layers with frequency disorder determine the nucleation in the other layers with uniform frequency distribution, regulating their phase transitions. Notably, the intermediate range of interlayer coupling strength significantly lowers the intralayer coupling strength required for synchronization. In particular, we find that desynchronization transitions are manifested due to both abrupt single-step and gradual multistep transitions in an intermediate range of the interlayer coupling strength, corroborating that such desynchronization phase transitions are specific to multilayer networks. We elucidate the distinct synchronization transitions resulting from heterogeneous nucleation using synchronization index and snapshots of coupling weights. Further, we deduce the evolution equations by reducing the full model to cluster level description using the collective coordinate framework. The transitions observed in the reduced model accurately capture those observed in the full model.