Role of contrarians in Kuramoto phase oscillator networks

Abstract

Synchronization of networked phase oscillators has proven itself to be an important process in understanding the collective behavior of a variety of real-world complex systems ranging from physical to biological systems (1; 2). Gardeñes et al. (3) reported that the transition to synchronization can be an abrupt or first-order type, called explosive synchronization (ES), and can be achieved by setting a correlation between the natural frequencies and respective degrees of networked phase oscillators. Owing to the significance of ES in elucidating various abrupt transitions found in real-world systems, for instance, massive blackouts (cascading failure of the power stations) (4) or epileptic seizure (the abrupt synchronous firing of neurons) (5), chronic pain in the FM brain (6), and bistable Cdc2-cyclin B in embryonic cell cycle (7), ES has received tremendous attention from the network science community (8; 9; 10). Experimental evidence of explosive synchronization has also been reported in star networks of electronic circuits (11) and mercury beating-heart oscillators (12). Tanaka et al. (13) introduced the occurrence of the first-order (discontinuous) transition resulting from finite inertia in the networked Kuramoto oscillators. Lately, further studies on ES have demonstrated that the microscopic correlation between degree frequency is not the only criterion for the occurrence of ES. For instance, ES is also shown to result from a fraction of adaptively controlled oscillators (14; 15; 16), and by assuming a positive correlation between coupling strengths and respective absolute of natural frequencies of networked oscillators (17) in isolated networks. All the investigations reinforce the fact that any suppressing factor, that hampers the growth of the largest synchronized clusters, leads to the abrupt formation of a single giant synchronous component. Lately, the investigations on ES have been extended to multilayered networks by considering a fraction of adaptively controlled Kuramoto oscillators, second-order Kuramoto oscillators (with inertia) (18), and intertwined multilayer couplings.