EXPLOSIVE SYNCHRONIZATION OF COMBINATIONAL PHASES ON RANDOM MULTIPLEX NETWORKS

The coherent dynamics of a large ensemble of interconnected dynamical units can be characterized by the synchronization process of coupled oscillators. In many real situations, each unit, in the meantime, may exist in multilayer networks, where the composite state of the unit can be determined by the corresponding states on each layer. In this paper, a combinational phase is introduced to describe the joint action of several phases. The combinational phase is a linear superposition of the phase in each layer with a coupling parameter, in the same manner as the generation of voltage from electricity and resistance when applying the phaser method. We study the dynamics of combinational phases by applying the Kuramoto model on multiplex networks, in which the weight of each layer affecting the combinational phase is controlled by a coupling parameter. An abrupt transition is found to emerge in the synchronization of combinational phases by adjusting the coupling parameter. We also show that phases of oscillators in each single layer remain incoherent while the combinational ones are fully synchronized. Theoretical analysis of this explosive transition is studied on a multiplex network, of which one layer is a star network, and the other is a fully connected one. Our findings provide a first understanding of the explosive critical phenomena of combinational phases on multiplex networks.